Consider a drum head constructed by stretching a membrane over a stiff frame that encloses a flat 2D domain. The vibration of the membrane is described by the wave equation (Helmholtz equation) with the Dirichlet boundary condition at the periphery of the domain where the membrane is constrained by the stiff frame. In this case, there is a set of discrete solutions to the wave equation, called *normal modes* or *eigenmodes*, each of which vibrates at a characteristic frequency, called *eigenfrequencies*.

The lowest eigenfrequency defines the fundamental tone, which for instance could be concert pitch A (440 Hz). The set of higher eigenfrequencies, or *overtones* in musical terms, gives rise to the tone color or timbre of the vibrating membrane. Kac’s lecture drew our attention to the eigenfrequencies: Is it possible to construct two drum heads with different shapes that share a set of eigenfrequencies? The idea was that if the two drums have an identical set of eigenfrequencies (being *isospectral*), then they would have the same timbre and sound the same to the ear, even though their shapes are different.

Kac commented on the asymptotic behavior of the eigenfrequencies in the limit of very high frequencies and made connections to various branches of physics and mathematics to provide a ground for intuitive understanding. The uniqueness question (in 2D flat space) remained unsolved until over two decades later when Gordon, Webb, and Wolpert finally constructed two polygons with an identical set of eigenvalues (see “One cannot hear the shape of a drum” and “Isospectral plane domains and surfaces via Riemannian orbifolds“).

The eigenvalues of the two polygons can be computed numerically, which is shown in this Isospectral Drums model in our Model Gallery.

The image below shows the first three normal modes of the two polygons that share the same set of eigenfrequencies:

In Gordon and Webb’s easy-to-read introductory article on this subject (“You can’t hear the shape of a drum”, *American Scientist*, vol. 84 (1996), pp. 46–55), they commented that such isospectral drums with different shapes are expected to be the exception, not the rule. In other words, they expected that, in general, one *can* hear the shape of a drum, unless the shape of the drum is specially constructed to be isospectral with another shape, like the two polygons depicted above.

In the following discussion, we will take a closer look at such special shapes by considering various physical mechanisms involved in the sound production and detection. We will find that when we include relevant physical effects, we actually *can* tell two drums apart by the sound, even if they are specially constructed to share the same set of eigenfrequencies.

The first effect we will examine is the excitation of the vibrational modes in the membrane. Since the timbre is determined by the set of relative amplitudes of the normal modes, it is not enough to just have an identical set of eigenfrequencies for the two drums to sound the same. They also need to have the same relative amplitude for each eigenmode, which may not be trivial to achieve.

Let’s take, for example, the same two polygonal drums from above and hit them with a drum stick at a few arbitrary places, one at a time, as such:

Each location of striking is somewhere in the middle of the drum, where a child may instinctively choose to hit if given such a drum and a drum stick. We use COMSOL Multiphysics simulation software to calculate the frequency response of each of the locations and plot the results in the graphs below.

We first focus on just one drum, say, the one on the left. Here is a plot showing the left drum’s frequency response:

As we hinted at earlier, the drum sounds differently depending on the location where it is struck by the drum stick. We see different energy distribution among the first three eigenmodes, which will result in different timbre. This is, of course, a well-known fact to percussionists and is the result of the same principle that enables a single bell to ring in two distinct tones, as demonstrated by this ancient set of bells from over two thousand years ago.

Now we know we can’t even make one drum sound the same unless we have a perfect aim of the drum stick. Is there any hope that we can make the two different drums sound the same?

In the graph below, we’ve added the frequency response of the second drum (the dashed curves). As we examine the graph, it becomes evident that none of the dashed curves match the solid curves in all three of the eigenmodes. In other words, the two drums do sound differently, even though they are isospectral, sharing the same set of eigenfrequencies.

Of course, we haven’t done an exhaustive search of all the possible combinations of striking locations. However, this simple example illustrates that it is not an easy job to make the drums sound the same, due to the different coupling strengths of energy from the drum stick to the various vibrational modes of the membrane.

The magic of mathematics never ceases to amaze us. Not long after the two isospectral polygons were published, Buser, Conway, Doyle, and Semmler constructed a pair of domains that are not only isospectral (sharing the same set of eigenfrequencies), but also *homophonic*: having a special point in the domain such that “corresponding normalized Dirichlet eigenfunctions take equal values at the distinguished points” (“Some planar isospectral domains“). In other words, if the special point of each drum is hit by a drum stick, then each corresponding pair of eigenmodes of the two drums will be excited with the same amplitude and the two drums will sound the same.

Shown below are the first few normal mode shapes computed numerically:

The special point of each domain is marked with a blue square in the schematic below:

In the following graph, we plot the computed frequency response of the two drums to a narrow Gaussian area load centered on each of the special points:

Isn’t it amazing how well the two frequency response curves (solid blue curve and green circles) lie on top of each other? With such a perfectly matched vibrational energy spectrum, wouldn’t the two drums sound exactly the same? Let’s continue our journey to explore more physical effects and find out.

Our ears do not sense the vibration of the membrane directly. Rather, the sensing is mediated by the acoustic wave in the air. Let’s set up the two homophonic drums outdoors, where the sound is allowed to propagate away from the drums without significant reflection. In this case, we can easily compute the frequency spectrum of the sound wave using COMSOL Multiphysics to find out what we really hear with our ears.

Let’s take a look at the three vibrational modes with the highest energies at about 111, 146, and 184 Hz as shown in the spectral graph above. For convenience, we will call them the first, second, and third mode, with the understanding that there are other modes in between being neglected since they are much less energetic.

The polar graph below compares the computed sound pressure level (in dB) in the plane of each of the two drums, a few meters away from each drum.

We see that the sound pressure field produced by the first mode is more or less independent of direction (solid and dashed blue curves). This is not surprising, since the mode shape of each drum looks pretty much like a monopole source:

On the other hand, the directionality of the sound field from the second or the third mode of each of the drums is quite pronounced and also quite different between the two drums. For example, for the second mode, the sound field from Drum 1 looks like a dipole field (solid red curve), while the one from Drum 2 is more complex (dashed red curve). This observation again matches what we see in the mode shapes of the two drums:

What really determines the perceived timbre is the ratio of the amplitudes of the higher modes (the overtones) to the lowest mode (the fundamental tone). So, in the next graph, we plot the amplitude ratios of the second and the third modes to the first mode, at a sampling of directions:

The blue square points are from Drum 1 and the red round points from Drum 2. The graph can be viewed as a map of timbre — if two points on the graph are near each other, then they sound similar; on the other hand, if two points on the graph are far away from each other, then they have very distinct timbre. As qualitatively illustrated by the green dashed boundaries, each drum can produce a range of timbre that the other cannot, in some range of directions.

As long as a listener is allowed to move around each drum, perhaps blindfolded, he or she will hear distinct ranges of timbre that tell the two drums apart. Therefore, even though the two “homophonic” drums share the same energy spectrum in their vibration modes, due to the difference in the mode shape and to the difference in energy transfer to the sound field in the air, the acoustic energy spectrum in some range of directions can be quite different. This is what would cause the two drums to sound differently to our ears.

In the previous analysis, we ignored the reaction force acted on the membrane by the air, the so-called *air loading effect*. It turns out that this effect is very significant for a real drum, since, after all, the entire area of the membrane participates in the pushing and pulling of the air around it.

We can simulate this effect using the *Acoustic-Structure Boundary* Multiphysics coupling feature of COMSOL Multiphysics. We find, for example, that the eigenfrequency of the second mode that we were discussing shifts from 146 Hz down to about 86 Hz. In addition, the magnitudes of shift of the two drums are different. The eigenfrequency of one drum was shifted down to 85.6 Hz, while the one of the other drum shifted to 86.8 Hz. This difference causes a pitch difference of about 23 cents, which is very audible in a side-by-side comparison.

Therefore, not only do the two drums differ in timbre (in some range of directions), they also differ in absolute pitch when we take the air loading effect into account.

The graph below shows the frequency response of the two drums around this mode. The difference in the resonant frequency is clearly seen, as well as the difference in the width of the resonance. There should be no doubt in our mind that with such different frequency responses, the two drums will produce easily distinguishable sounds.

It is a great achievement in mathematics to invent the isospectral drums that share the same set of eigenfrequencies and the homophonic drums that share the same power spectrum of the vibrational modes when excited at a special point. However, these phenomena only happen in vacuum, where there is no sound. Once we put the drums to test in air, they start to sound differently due to the air loading effect and the directionality of the energy transfer from the membrane to the sound wave.

In his lecture, Kac told the early 20^{th}-century story of Lorentz calling for mathematicians’ attention to the eigenvalue problem involved in the theory of black body radiation and Weyl answering the call with the proof of the theorem of the asymptotic behavior of eigenvalues at very high frequencies.

Here, we could use the help of our mathematician friends again, even though the subject matter may not be as grand as black body radiation and quantum mechanics. Is it possible to construct homophonic drums with different shapes that sound the same when including directionality and air loading effects? It may be possible to pose this as an optimization problem to solve numerically for a solution with a finite set of audible frequencies.

However, the computation cost will be high and the result will be approximate. An elegant analytical solution similar to those shown in the papers mentioned above would be much nicer. I hope this will arouse the interest of mathematicians who are reading.

]]>When inside a room — a conference room, concert hall, or even a car — everyone has an opinion of when the “acoustics” are good or bad. In *room acoustics*, we want to study this notion of sound quality in a quantitative way. In short, room acoustics is concerned with assessing the acoustics of enclosed spaces. The Acoustics Module of COMSOL Multiphysics has several tools to simulate the acoustics of rooms and other confined spaces. I will present those here.

When sound is emitted inside a room, a listener will perceive the sound as a combination of direct sound from the source as well as sound reflected off the walls. At the walls, the sound is reflected, absorbed, and scattered.

Since all of these processes are frequency dependent, a poorly designed meeting room can, for example, be highly reverberant in a frequency band that is important for speech. The room could also have a strong modal behavior (standing waves) at certain critical frequencies that are easily excited. These are things you want to avoid and be able to predict when designing a room.

Architects and civil engineers want to control the sound field by placing absorbers, diffusers, and reflectors in appropriate locations. In concert halls, you want to maximize the listening experience where the audience is located. In office spaces, you want to avoid anything that can seem noisy and disturb the concentration of employees. In classrooms and lecture halls, you want to ensure clear perception of speech. The sound environment is important for various reasons, which is why there are national standards and regulations for the sound environment in many cases.

Refurbishing a badly designed room can be very expensive, so you do not want to rely only on measurements on scale models or measurements done after the fact. Modeling the room acoustic behavior beforehand is important and essential in order to optimize and perfect the design. Simulation models and measurements need to relate architectural aspects (geometry) to subjective observations using physical measures (metrics). This is done by calculating a long range of room acoustic measures, such as the reverberation time, early decay time, clarity, and many other standardized parameters.

The modeling approach you want to adopt depends on the studied frequency (the wavelength compared to geometrical features of the room). In the Acoustics Module of the COMSOL suite of FEA software, we essentially offer three approaches packaged in three physics interfaces. The *Pressure Acoustics* interface can model the modal behavior in rooms. The *Ray Acoustics* interface and the *Acoustic Diffusion Equation* interface cover the high frequency limit or reverberant behavior (geometrical acoustics). I discuss the interfaces and their applicability in the sections below.

*Animation of the ray front position as they are released inside a small concert hall. The color scale gives an impression of the ray intensity on a logarithmic scale.*

As mentioned above, room acoustics is typically divided into three categories, depending on the studied frequency. Or, more specifically, depending on the wavelength compared to the characteristic geometric features of the room in question.

In the low-frequency range, the room resonances dominate. This is known as the *modal region*. At the other end of the scale, in the high-frequency limit, the wavelength becomes smaller than the characteristic geometrical features of the room. Here, we deal with the reverberant region or the *geometrical acoustics limit*. Between the modal and the high-frequency limit, there is a so-called *transition zone*. Note that there is no clear-cut definition of this zone.

Classical room acoustics theory provides some tools that enable a back-of-the-envelope assessment of the behavior of a room. For a given room, the Schroeder frequency, f_\textrm{s}, predicts the limiting frequency between the modal behavior and the high-frequency reverberant behavior of the room.

The Schroeder frequency is given by:

(1)

f_\textrm{s} = 2000 (\textrm{m}/\textrm{s})^{3/2} \sqrt{\frac{T_{60}}{V}}

where V is the room volume and T_{60} is the reverberation time.

The equation is based on the criterion (suggested by Schroeder) that at the limit, three eigenfrequencies fall into one resonance half-width. The reverberation time (or decay time), T_{60}, is the time required for the sound pressure level (created by an impulse source) to decay 60 dB. A first simple approximate measure of the reverberation time is given by the well-established Sabine formula:

(2)

T_{60} = \frac{55.3 V}{c A}, \qquad A = \Sigma S_i \alpha_i

Here, c is the speed of sound and A is the total absorption, where S_i and \alpha_i are the surface area and absorption of the i^{th} surface, respectively.

This is possibly the best-known formula in room acoustics. The equation stems from a classical statistical room acoustics analysis assuming a pure diffuse sound field. In a diffuse sound field, the sound pressure level is uniform and the reflected sound dominates. This phenomenon is also known as a *reverberant sound field*. In such a field, the damping constant (related to the overall absorption) can be approximated and relates to the reverberation time.

The modal behavior of rooms and enclosed spaces is best analyzed solving Helmholtz equation or the scalar wave equation using the finite element method. In the reverberant or high-frequency limit at frequencies above the Schroeder frequency, you may utilize two different approaches. Your choice depends on the assumptions that can be made and the desired level of detail.

The *Acoustic Diffusion Equation* interface may be used in the purely diffuse sound field limit, neglecting all direct sound. This is a fast method to assess reverberation times and sound pressure level distributions in systems of coupled rooms. The ray tracing capabilities of the *Ray Acoustics* interface provide a much more detailed picture including the direct sound and early reflections. With this interface, you also have the ability to reconstruct an impulse response.

Up to the Schroeder frequency, the modal behavior of rooms is important, where standing waves dominate over the reverberant nature. Inside a car, the transition may be as high as somewhere between several hundreds of Hertz up to 1000 Hz. In a small office, it may be up to 200 Hz, while in large concert halls, the transition is typically below 50 Hz. In the small concert hall model shown below, the Schroeder frequency is 115 Hz (the reverberation time is about 1.3 s and the volume is 430 m^{3}). The modal behavior is important for subwoofer systems in cinemas, for instance.

The modal behavior as well as the room eigenfrequencies are best analyzed using the *Pressure Acoustics* interface. A frequency domain study can reproduce a transfer function for the bass system. You can also use it to analyze dead regions or find eigenfrequencies. A transient study is interesting when, for example, looking at bass build-up transients inside a car cabin.

Models of interest here are:

*Pressure distribution for the first eigenmode inside a small room. From the Eigenmodes of a Room model.*

If you want to compute the trajectories, phase, and intensity of acoustic rays, you should choose the *Ray Acoustics* interface. Ray acoustics is a good choice when working in the high-frequency limit where you have an acoustic wavelength that is smaller than the characteristic geometric features. The interface is not limited to modeling acoustics in closed spaces, like rooms and concert halls, but can also be used in outdoor environments. At exterior boundaries, you can assign various wall conditions, such as combinations of specular and diffuse reflections. The frequency, intensity, and direction of the incident rays may influence both impedance and absorption.

Below are two figures from the Small Concert Hall model found in the Model Gallery for the Acoustics Module.

The figure to the left depicts the ray paths for a selected number of rays emitted from a source located on the small stage. The figure to the right depicts the energy response as measured in the center of the room. The dots represent the simulated ray response (5,000 rays are released) and the green and red curves represent decay curves based on simple Sabine-like estimates of the reverberation time T_{60}. The cyan curve is a so-called *Schroeder integration* of the energy response, yielding the energy-decay curve. All four agree well when the response is measured in the center of the room.

*Left: Ray path for a selected number of rays emitted from a source located on a small stage. Right: The energy impulse response compared with two simple decay measures and the energy decay curve.*

With the *Ray Acoustics* interface, the response can be measured at any point in the concert hall. The properties of absorbers and diffusers can be both frequency-dependent and angle-of-incidence dependent. Thus, the listening environment can be well described, analyzed, and optimized. The simple estimates are not accurate everywhere in a room and not for complex room geometries.

The *Acoustic Diffusion Equation* interface solves a diffusion equation for the acoustic energy density distribution for room acoustics. The method is also sometimes referred to as *energy finite elements*. This method is an extension of the principles used to calculate the Sabine reverberation time in Equation 2. This particular interface is applicable for high-frequency acoustics when the acoustic fields are diffuse. The diffusion of the acoustic energy density depends on the mean free acoustic path and thus on the room geometry. Absorption may be applied at walls and a transmission loss may be applied when coupling rooms. Increased diffusion due to room fitting can be added. Material properties and sources may be specified in frequency bands.

Compared to a ray acoustics simulation, this interface does not include any phase information, direct sound, or early reflections. The interface supports stationary studies for modeling a steady-state sound energy or sound pressure level distribution. You can use a time-dependent study to determine energy decay curves and reverberation times. You can use an eigenvalue study to determine the reverberation time of coupled and uncoupled rooms. The eigenvalue is directly related to the exponential decay time and so the reverberation time.

We utilized all three study types in the One-Family House Acoustics model, which studies the acoustics in a single-family home with a noise source located in the living room.

*Energy flux and SPL distribution inside a two-story single-family house.*

Check back on the COMSOL blog this spring for specific blog posts about the *Acoustic Diffusion Equation* and *Ray Acoustics* physics interfaces.

In the meantime, here is a list of suggested reading material:

- H. Kuttruff,
*Room Acoustics*, CRC Press, Fifth Edition, 2009. - A. D. Pierce,
*Acoustics, An Introduction to its Physical Principles and Applications*, Acoustical Society of America, 1991. - ISO 3382 Standard, Measurement of room acoustic parameters.
- M. R. Schroeder,
*New Method of Measuring Reverberation Time*, J. Acoust. Soc. Am., 37 (1965). - M. R. Schroeder,
*Integrated-Impulse method measuring sound decay without using impulses*, J. Acoust. Soc. Am., 66 (1979).

Acoustic radiation force is an important nonlinear acoustic phenomenon that manifests itself as a nonzero force exerted by acoustic fields on particles. Acoustic radiation is an acoustophoretic phenomenon, that is, the movement of objects by sound. One interesting example of this force in action is the acoustic particle levitation discussed in this previous blog post. Today, we shall examine the nature of this force and show how it can be computed using COMSOL Multiphysics.

To understand the nature of the acoustic radiation force, let’s first consider a simple example of a particle in a standing wave pressure field (here assumed to be lossless).

The force on the particle arises as a result of particle’s finite size, such that the gradients in the pressure field will result in greater force being exerted on one side of the particle than the other. However, if we are considering a harmonic pressure wave, then the force is expected to behave as a harmonic function, which can be expressed as F_\text{harmonic} = F_0\sin (2\pi f_0 t+\phi). I’ve shown this as a black arrow in the animation below.

If time-averaged, the total contribution goes to zero. So, where does the observed nonzero force come from?

This question was first addressed by L. V. King back in 1934 (“On the acoustic radiation pressure on spheres“). In order to understand King’s results, we must take a step back to examine how the governing equations of acoustics are derived.

We will find out that they emerge from the Navier-Stokes equations as a result of a linearization procedure, which is normally carried out in two steps.

First, a very small time-varying perturbation in pressure and velocity is assumed on top of a stationary background field. When time derivatives are applied, the stationary terms drop out and what is left only includes the time-dependent perturbation terms. The remaining expression will contain both linear and nonlinear contributions. The latter appear in the form of products of two or more linear perturbation terms, and they result from convective and inertial terms in the original Navier-Stokes equation.

But, in the simplest acoustic limit, the contribution of nonlinear terms can be neglected because the amplitudes of perturbations considered are very small. For example, 0.01^{2} is much smaller than 0.01 and can therefore be neglected. So, in the second step of the linearization procedure, all the nonlinear terms are neglected and the linear wave equation is obtained.

What King has indicated was that in order to understand and evaluate the effect of acoustic radiation force, the nonlinear terms must somehow be retained in the equations.

Keeping the terms up to a second order, the pressure field will appear as a combination of two terms p = p_1 + p_2, where p_1 and p_2 can be expressed in a simplistic form as p_1 = \rho_0 c_0 v, which appears as a linear function of the perturbation velocity v, and p_2 = 1/2 \rho_0 v^2, which appears as a nonlinear function of v. Since, in the acoustic limit, we only consider the cases in which v \ll c_0, where c_0 is the adiabatic speed of sound, we conclude that p_2 \ll p_1.

At this point, we are ready to answer the first question: Where does the acoustic radiation force come from?

Going back to the example of a particle in a standing wave pressure field, let’s examine the linear and nonlinear components of the pressure and the forces produced by these components. In this case, p_1 will be a time-harmonic function p_1 = P_1 \cos(kx)\sin(2\pi f t) and p_2 will be an an-harmonic function p_2 = P_2\cos^2(kx)[1-cos(4\pi f t)] resulting from the nonlinear contribution.

These terms are visualized by the waveforms in the animation above. The forces resulting from these pressure terms are indicated by arrows. The linear force (black arrow) changes both in magnitude and direction so its cycle-averaged contribution is zero, whereas the nonlinear term (red arrow) only changes in magnitude and on average exerts a nonzero force.

The simple analysis above demonstrates the main mechanism underlying the acoustic radiation force phenomenon. Intuitively, we realize that no force will appear if the particle has the same acoustic properties as the surrounding medium. In other words, the radiation force should be a function of not only the size of a particle and the amplitude of the acoustic field, but also of the particle’s *acoustic contrast* (the ratio of the material properties of the particle relative to the surrounding fluid).

Due to the acoustic contrast, the field incident on the particle will be reflected from its surface and the radiation force will be a result of a combination of incident and reflected waves. This makes the problem quite difficult to solve analytically. The solution in a closed analytic form was only given for some limiting cases by a number of authors, starting with King. He has considered rigid spherical particles with dimensions much smaller than the wavelength of the incident wave, but much larger than the viscous and thermal skin depths. It was the second assumption that allowed these terms to be neglected.

King’s results have been extended to include compressible particles as in “Acoustic radiation pressure on a compressible sphere“. The results from this study were later confirmed by L. P. Gor’kov in 1962 in “On the forces acting on a small particle in an acoustical field in an ideal fluid”. Viscous and thermal effects become important when the size of the particles becomes comparable to the acoustic boundary layers (thermal and viscous). Results including viscosity were recently presented in 2012 by M. Settnes and H. Bruus.

Gor’kov has developed an elegant approach to expressing the radiation force in terms of time-averaged kinetic and potential energies of stationary acoustic fields of any geometries. His results, when applied to small compressible fluid particles, give the force as a gradient of a potential function U_\text{rad}:

(1)

\mathbf{F} = -\nabla U_\text{rad},

The potential function U_\text{rad} is expressed using the acoustic pressure and velocity as:

(2)

U_\text{rad} = V_p \left [ f_1\frac{1}{2\rho c^2}\langle p^2 \rangle - f_2\frac{3}{4}\rho\langle v^2 \rangle\right],

where V_p is the volume of the particle and the scattering coefficients are given by:

(3)

f_1 = 1 -\frac{K_0}{K_p},\ \ \ \ \ f_2 = \frac{2(\rho_p-\rho)}{2\rho_p+\rho},

where K_i are the bulk moduli. The scattering coefficients f_1 and f_2 represent the monopole and dipole coefficients, respectively. This approach, which is based on the scattering theory, is only valid for particles that are small compared to the wavelength \lambda in the limit a/\lambda \ll 1, where a is the radius of the particle.

The v and p terms that appear in Eq. (1) are the first-order terms that can be obtained by solving a linear acoustic problem. Results in this form are typically obtained using a perturbation method, which is widely practiced in physics. A thorough review and examples of this method applied to nonlinear problems in acoustics and microfluidics can be found in a textbook by Professor Henrik Bruus titled *Theoretical Microfluidics*.

Eq. (1) is coded in the COMSOL Multiphysics *Particle Tracing for Fluid Flow* interface to evaluate the acoustic radiation force on particles. But, as mentioned above, it only applies to acoustically small particles and neglects thermoviscous effects. An example can be seen in the Acoustic Levitator model. Knowing the radiation force is important when modeling and simulating systems that handle particles using this phenomenon. This can be, for instance, microfuidic systems that sort and handle cells and other particles. An example of this is discussed in the blog post Acoustofluidic Multiphysics Problem: Microparticle Acoustophoresis.

To extend the theory beyond the limit of acoustically small particles, a numerical approach is required. We will consider that next.

In general, all forces can be expressed using momentum fluxes as \mathbf F = \int_S T \mathbf{n} d\mathbf{a}, where the surface of integration, S, is the external surface of the particle.

Gor’kov has used this fact to obtain a closed-form analytical expression for a force acting on a particle in an arbitrary acoustic field. To compute the nonlinear acoustic radiation force, the momentum flux due to the acoustic field has to be evaluated up to second-order terms. The main appeal of his result is that, as mentioned earlier, the second-order terms can be expressed using the solution of a linear problem.

To implement his method, all we need to do is solve the acoustic problem, use the results to compute the second-order momentum flux, and substitute the solution into the flux integral.

H. Bruus has shown that neglecting the thermoviscous effects, the second-order flux terms are:

(4)

T = -\frac{1}{2\rho c^2}\langle p^2 \rangle + \frac{1}{2}\rho\langle v^2 \rangle

The integral should be taken over a surface of a particle moving in response to the applied force. This means that the surface of integration is a function of time S = S(t). To overcome this difficulty, Yosioka and Kawasima have indicated that the integration can be transformed to an equilibrium surface S_0 that encloses the particle. Compensating for the error with the addition of a convective momentum flux term, the force, in total, becomes:

(5)

\mathbf F = \int_{S_0} T \mathbf{n} d\mathbf{a} -\int_{S_0}\rho \langle(\mathbf{vn})\cdot \mathbf{v}\rangle d\mathbf a

All that is left to do now is solve the acoustics problem to obtain the acoustic pressure and velocity and substitute them into the integral in Eq. (5). In contrast to the approach used in Eq. (1) to (3), the force expression given in Eq. (5) is valid for all particle sizes as long as the stress T is given. This approach was recently implemented in COMSOL Multiphysics by a group of researchers from the University of Southampton.

It should be noted that the expression in Eq. (4) is only true when the viscous and thermal effects are neglected. If these losses are included the integration surface S_0 should be taken outside of the boundary layers or a correct full stress expression for T used on the particle surface. A first principle perturbation approach including thermal and viscous losses was presented at the 2013 ICA-ASA conference by M. J. Herring Jensen and H. Bruus, titled “First-principle simulation of the acoustic radiation force on microparticles in ultrasonic standing waves”. A detailed derivation of the governing equations up to second order, in a form suited for implementation in COMSOL Multiphysics, is given in the paper “Numerical study of thermoviscous effects in ultrasound-induced acoustic streaming in microchannels“.

To benchmark the method presnted by Glynne-Jones et al., let’s compute an acoustic radiation force exerted by a standing wave on a spherical nylon particle immersed in water. We assume a frequency of 1 MHz and a pressure amplitude of 1 bar and implement the model using the *Acoustic-Structure Interaction* interface in 2D axisymmetric geometry. The size of the box in the model is four wavelengths high and two wavelengths wide.

Let’s excite a standing wave in this box using a Background Pressure Field condition, set up in such way that the particle is at a distance of \lambda/8 from the pressure node.

The integrals in Eq. (5) are computed by setting up integration coupling operators in the *Component 1 > Definitions* node. We need to make sure that the integral is calculated in the revolved geometry by checking the appropriate box and selecting the boundaries of the particle to define the surface of integration.

It is noteworthy to mention here that the force computation used in this method is independent of the surface of integration due to the conservation of flux as long, as it is located outside the particle. In fact, using a surface at larger distances will be more numerically accurate, simply because there will be more points to use for numerical evaluation of the integral. To perform this integration, we can add another surface external to the particle.

Finally, new flux variables are introduced in the *Component 1 > Definitions as Variables 1a* node. They are used as arguments for the integration operators to compute the total force.

We are now ready to compare the perturbation approach to an analytical solution.

As expected, they compare reasonably well for small particle radii where the analytical solution considered is valid. Some analytical models that include higher harmonics in scattered field decomposition offer solutions that agree with the outlined numerical approach for large and small spherical particles (as in the paper by T. Hasegawa, “Comparison of two solutions for acoustic radiation pressure on a sphere“.)

A small discrepancy for small particle radii between analytical and numerical methods may be attributed to the fact that the theoretical models assume that the particle is plastic, whereas in this example, we have considered an elastic particle with bulk modulus of 0.4.

The perturbation method has a number of advantages.

First, it exploits the linear acoustics method to evaluate nonlinear second-order force effects. This allows the analysis to be easily extended to 3D for particles of arbitrary shapes and material composition. For example, we can extend it to simulate acoustic radiation forces on biological cells or microbubbles.

Second, because the acoustic equations are solved in the frequency domain where very efficient numerical methods are well established, the solution time in COMSOL Multiphysics is quite fast even in 3D.

Meanwhile, the disadvantage of this method is that it is driven by theoretical results that rely on a set of simplifying assumptions, and it can only be validated in a limited number of cases. What we would like to have is a numerical method that allows the problem to be solved directly.

We shall see how this can be achieved in the next blog post. Stay tuned!

- Blog posts:
- Model: Acoustic Levitator
- L. V. King, “On the acoustic radiation pressure on spheres“, (1934)
- M. Settnes and H. Bruus, “Forces acting on a small particle in an acoustical field in a viscous fluid“, Phys. Rev. E 85, 016327 1-12 (2012)
- P. Glynne-Jones, P. P. Mishra, R. J. Boltryk, and M. Hill, “Efficient finite element modeling of radiation forces on elastic particles of arbitrary size and geometry“, J. Acoust. Soc. Am., 133, 1885-93 (2013)
- P.B. Muller, R. Barnkob, M.J.H. Jensen, and H. Bruus, “A numerical study of microparticle acoustophoresis driven by acoustic radiation forces and streaming-induced drag forces“, Lab Chip 12, 4617-4627 (2012)
- P.B. Muller and H. Bruus, “Numerical study of thermoviscous effects in ultrasound-induced acoustic streaming in microchannels“, Phys. Rev. E 90, 043016 1-12 (2014)

The piezoelectric modeling interface seeks to:

- Make the modeling workflow more
- Transparent
- Flexible

- Enable you to debug the models more easily

This will allow you to successfully simulate piezoelectric devices as well as easily extend the simulation by coupling it with any other physics.

You may already be familiar with the three different modules that can be used for simulating piezoelectric materials:

Each of these modules gives you a predefined *Piezoelectric Devices* interface that you can use for modeling systems that include both piezoelectric and other structural materials. The Acoustics Module offers two predefined interfaces, namely the *Acoustic-Piezoelectric Interaction, Frequency Domain* interface and the *Acoustic-Piezoelectric Interaction, Transient* interface. These two allow you to model how piezoelectric acoustic transducers interact with the fluid media surrounding them.

*The *Piezoelectric Devices* interface is available in the list of structural mechanics physics interfaces.*

*The *Acoustic-Piezoelectric Interaction, Frequency Domain *and the* Acoustic-Piezoelectric Interaction, Transient* interfaces are available in the list of acoustics physics interfaces.*

These predefined multiphysics interfaces couple the relevant physics governing equations via constitutive laws or boundary conditions. Thus, they offer a good starting point for setting up more complex multiphysics problems involving piezoelectric materials. The new piezoelectric interfaces in COMSOL Multiphysics version 5.0 provide a transparent workflow to visualize the constituent physics interfaces. There is also a separate Multiphysics node that lists how the constituent physics interfaces are connected to each other.

Let’s find out how these multiphysics interfaces are structured.

Upon selecting the *Piezoelectric Devices* multiphysics interface, you see the constituent physics: *Solid Mechanics* and *Electrostatics*. You also see the *Piezoelectric Effect* branch listed under the Multiphysics node, which controls the connection between *Solid Mechanics* and *Electrostatics*.

*Part of the model tree showing the physics interfaces and multiphysics couplings that appear upon selecting the* Piezoelectric Devices *interface.*

By default, all modeling domains are assumed to be made of piezoelectric material. If that is not the case, you can deselect the non-piezo structural domains from the branch *Solid Mechanics > Piezoelectric Material*. These domains then get automatically assigned to the *Solid Mechanics > Linear Elastic Material* branch. This process ensures that all parts of the geometry are marked as either piezoelectric or non-piezo structural materials and that nothing is accidentally left undefined.

If you are working with other material models that are available with the Nonlinear Structural Materials Module, such as hyperelasticity, you can add that as a branch under *Solid Mechanics* and assign the relevant parts of your modeling geometry to this branch. The Solid Mechanics node gives us full flexibility to set up a model that involves not only piezoelectric material but also linear and nonlinear structural materials. The best part is that if these materials are geometrically touching each other, the COMSOL software will automatically take care of displacement compatibility across them.

If some parts of the model are not solid at all, like an air gap, you can deselect them in the Solid Mechanics node.

From the Solid Mechanics node, you will also assign any sort of mechanical loads and constraints to the model.

The Electrostatics node allows you to group together all the information related to electrical inputs to the model. This would include, for example, any electrical boundary conditions such as voltage and charge sources. By default, any geometric domain that has been assigned to the *Solid Mechanics > Piezoelectric Material* branch also gets assigned to the *Electrostatics > Charge Conservation, Piezoelectric* branch. If you have any other dielectric materials in the model that are not piezoelectric, you could assign them to the *Electrostatics > Charge Conservation* branch.

The *Multiphysics > Piezoelectric Effect* branch ensures that the structural and electrostatics equations are solved in a coupled fashion within the domains that are assigned to the *Solid Mechanics > Piezoelectric Material* (and also the *Electrostatics > Charge Conservation, Piezoelectric*) branch.

The multiphysics coupling is implemented using the well-known coupled constitutive law for piezoelectric materials. Note that the *Electrostatics > Charge Conservation, Piezoelectric* branch is mainly used as a placeholder for assigning geometric domains that belong to the piezoelectric material model. This helps the *Multiphysics > Piezoelectric Effect* branch understand whether a domain assigned to the *Electrostatics* interface is piezoelectric or not.

Note: For an example of working with the

Piezoelectric Devicesinterface, check out the tutorial on modeling a Piezoelectric Shear Actuated Beam.

It is also possible to add effects of damping or other material losses in dynamic simulations. You can do so by adding one or more of the following subnodes under the *Solid Mechanics > Piezoelectric Material* branch:

*Damping and losses that can be added to a piezoelectric material.*

Subnode Name | When to Use the Subnode |
---|---|

Mechanical Damping | Allows you to add purely structural damping. Choose between using Loss Factor (in frequency domain) or Rayleigh damping (for both frequency and time domains) models. |

Coupling Loss | Allows you to add electromechanical coupling loss. Choose between using Loss Factor (for frequency domain) or Rayleigh damping (for both frequency and time domains) models. |

Dielectric Loss | Allows you to add dielectric or polarization loss. Choose between using Loss Factor (for frequency domain) and Dispersion (for both frequency and time domains) models. |

Conduction Loss (Time-Harmonic) | Allows you to add electrical energy dissipation due to electrical resistance in a harmonically vibrating piezoelectric material (for frequency domain only). |

Note: For an example of adding damping to piezoelectric models, check out the tutorial on modeling a Thin Film BAW Composite Resonator.

Additional damping also takes place due to the interaction between a piezoelectric device and its surroundings. This can be modeled in greater details using the Acoustic-Piezoelectric Interaction interfaces.

Upon selecting one of the Acoustic-Piezoelectric Interaction interfaces, you see the constituent physics: *Pressure Acoustics*, *Solid Mechanics* and *Electrostatics*. You also see the *Acoustic-Structure Boundary* and *Piezoelectric Effect* branches listed under the Multiphysics node.

*Part of the model tree showing the physics interfaces and multiphysics couplings that appear when selecting the *Acoustic-Piezoelectric Interaction, Frequency Domain* and the* Acoustic-Piezoelectric Interaction, Transient* interfaces.*

By default, all modeling domains are assigned to the *Pressure Acoustics* interface as well as the *Solid Mechanics > Piezoelectric Material* and* Electrostatics > Charge Conservation, Piezoelectric* branches. Note that the *Pressure Acoustics* interface is designed to simulate acoustic waves propagating in fluid media.

Since COMSOL Multiphysics cannot know a *priori* which parts of the modeling geometry belong to the fluid domain and which ones are solids, you are expected to provide that information by deselecting the solid domains from the *Pressure Acoustics, Frequency Domain* (or *Pressure Acoustics, Transient*) branch and deselecting the fluid domains from the *Solid Mechanics* and *Electrostatics* branches.

Once you do that, the boundaries at the interface between the solid and fluid domains are detected and assigned to the *Multiphysics > Acoustic-Structure Boundary* branch. This branch controls the coupling between the *Pressure Acoustics* and *Solid Mechanics* physics interfaces. It does so by considering the acoustic pressure of the fluid to be acting as a mechanical load on the solid surfaces, while the component of the acceleration vector that is normal (perpendicular) to the same surfaces acts as a sound source that produces pressure waves in the fluid.

Note: For an example of Acoustic-Piezoelectric Interaction, check out the tutorial on modeling a Tonpilz Transducer.

The transparency in the workflow as discussed above also paves the way for adding more physics and creating your own multiphysics couplings.

For example, let’s say there is some heat source within your piezoelectric device that produces nonuniform temperature distribution within the device. In order to model this, you can add another physics interface called *Heat Transfer in Solids* in the model tree and prescribe appropriate heat sources and sinks to find out the temperature profile. You could then add a *Thermal Expansion* branch under the Multiphysics node to compute additional strains in different parts of the device as a result of the temperature variation.

The *Multiphysics > Thermal Expansion* branch couples the *Heat Transfer in Solids* and the *Solid Mechanics* interfaces. It might also be possible that the piezoelectric material properties have a temperature dependency. You could represent these properties as functions of temperature and let the *Multiphysics > Temperature Coupling* branch pass on the information related to temperature distribution in the modeling geometry to the *Solid Mechanics* or even the *Electrostatics* branches, thereby producing additional multiphysics couplings.

*Part of the model tree showing the physics interfaces and multiphysics couplings that you can use to combine piezoelectric modeling with thermal expansion and temperature-dependent material properties.*

Similar to adding more physics and multiphysics couplings, it is also possible to disable one or more multiphysics couplings — or even any of the physics interfaces shown in the model tree. This could be immensely helpful for debugging large and complex models.

*The model tree on the left shows a scenario where the Piezoelectric Effect multiphysics coupling is disabled. The model tree on the right shows a scenario where the* Electrostatics* physics interface is disabled.*

For example, you can disable the *Multiphysics > Piezoelectric Effect* branch and solve for the *Solid Mechanics* and *Electrostatics* physics interfaces in an uncoupled sense. You could also solve a model by disabling either the *Solid Mechanics* or the *Electrostatics* interface.

Running such case studies could help in evaluating how the device would respond to certain inputs if there were no piezoelectric material in place. This approach could also be used to evaluate equivalent structural stiffness or equivalent capacitance of the piezoelectric material.

You could also start by adding only one of the constituent physics, say *Solid Mechanics*, and after performing some initial structural analysis, go ahead and add the *Electrostatics* physics interface to the model tree once you are ready to add the effect of a piezoelectric material.

In that case, when you add the *Electrostatics* physics on top of the existing *Solid Mechanics* physics in the model tree, the COMSOL software will automatically add the Multiphysics node. From there, you can manually add the *Piezoelectric Effect* branch. Note that if you take this approach of adding the constituent physics interfaces and multiphysics effect manually, you would also have to manually add the piezoelectric modeling domains to the *Solid Mechanics > Piezoelectric Material*, the *Electrostatics > Charge Conservation, Piezoelectric*, and the *Multiphysics > Piezoelectric Effect* branches.

In a similar fashion, you can continue to add more physics interfaces and multiphysics couplings to your model based on your needs.

To learn more about modeling piezoelectric devices in the COMSOL software environment, you are encouraged to refer to these resources:

- Piezoelectric Features Overview
- Acoustics Module User’s Guide
- MEMS Module User’s Guide
- Structural Mechanics Module User’s Guide

In particle tracing and ray tracing simulations, we often need to use the particle or ray properties to change a variable that is defined on a set of domains or boundaries. For example, solid particles in a fluid might exert a significant force on the surrounding fluid, and they may also erode the surfaces they hit.

In previous blog posts, I’ve discussed two other cases in greater detail: divergence of an electron beam due to self-potential and thermal deformation of lenses in a high-powered laser system. Each of these phenomena can be modeled using Accumulators or the specialized features that are derived from them.

An Accumulator is a physics feature that communicates information from particles or rays to the underlying finite element mesh. For each Accumulator feature in a model, a corresponding dependent variable, called an *accumulated variable*, is declared. These accumulated variables can be defined either within a set of domains or on a set of boundaries, and they can represent any physical quantity, making them extremely flexible.

The Accumulator features can be added to any of the physics interfaces of the Particle Tracing Module. They can also be used in the *Geometrical Optics* interface, available with the Ray Optics Module, and the *Ray Acoustics* interface, available with the Acoustics Module.

Depending on the physics interface, more specialized versions of the Accumulator may be available for computing specific types of physical quantities. For example, the *Particle Tracing for Fluid Flow* interface includes a dedicated *Erosion* boundary condition that includes several built-in models for computing the rate of erosive wear on a surface.

The Accumulators can be divided into three broad categories, which function in the following ways:

- Accumulators on boundaries increment a variable defined on a boundary element whenever a particle hits it.
- Accumulators on domains project information from each particle to the mesh elements the particle passes through.
- Nonlocal accumulators communicate information from a particle’s current position to the location where it was originally released.

We will now investigate each of these varieties in greater detail.

When particles or rays strike a surface, they can affect that surface in a wide variety of ways. For example, a laser can cause a boundary to heat up, sediment particles can erode their surroundings, and sputtering can occur when high-velocity ions strike a wafer in a process chamber. All of these effects require the same basic modeling procedure; we define a variable on the boundary and change its value when particles or rays interact with the boundary.

To begin, let’s consider a simple case in which we want to count the number of times a boundary is hit. We first define a variable, called `rpd`

, for example, which can have a distinct value in every boundary mesh element. Initially, this variable is set to zero in all elements. Every time a particle hits a mesh element on this boundary, we would like to increment the variable on that element by 1.

The values of the accumulated variable on the boundary elements (illustrated as triangles) after one collision are shown below:

To implement this in COMSOL Multiphysics, we first set up the particle tracing model, then add a “Wall” node to the boundary for which we want to count collisions. In this case, let’s specify that particles are reflected at this surface by selecting the Bounce wall condition. We then add the Accumulator node as a subnode to this Wall.

The settings shown in the following screenshot cause the accumulated variable (called `rpb`

) to be incremented by 1 (the expression in the Source edit field) every time a particle hits the wall.

I have created an animation that demonstrates how the number of collisions with each boundary element is counted over the course of the study. Check it out:

By changing the expression in the Source edit field, it is possible to increment the accumulated variable using any combination of variables that exist on the particle and on the boundary. For example, the accumulated variable may increase by a different amount based on the velocity or mass of incoming particles. The dependent variable need not be dimensionless. In fact, it can represent any physical quantity.

In addition to the generic Accumulator subnode — which can represent anything — dedicated accumulator-based features are available in the different physics interfaces, including the following:

- In the
*Charged Particle Tracing*physics interface:*Etch*(Use this to model physical sputtering of a surface by energetic ions.)*Current Density**Heat Source**Surface Charge Density*

- In the
*Particle Tracing for Fluid Flow*physics interface:*Erosion*(For computing the total mass removed from the surface or the rate of erosive wear.)*Mass Deposition**Boundary Load**Mass Flux*

- In the
*Geometrical Optics*physics interface:*Deposited Ray Power*(For computing a boundary heat source using the power of incident rays.)

We may also want to transfer information from particles to all of the mesh elements they pass through, not just the boundary elements they touch. We can do so by adding an Accumulator node to the physics interface directly, instead of adding it as a subnode to a Wall or other boundary condition.

For example, we can use an Accumulator to reconstruct the number density of particles within a domain. This technique is used in a benchmark model of free molecular flow through an s-bend in which the *Free Molecular Flow* interface is used to compute the number density of molecules in a rarefied gas.

Here is the geometry of the s-bend:

The settings window for the Accumulator is shown below.

The expression in the Source edit field is a bit more complicated than in the previous case. The source term R is defined as

(1)

R = \frac{J_{\textrm{in}} L}{N_{p}}

where J_{\textrm{in}} (SI unit: 1/(m^2 s)) is the molecular flux at the inlet, L (SI unit: m) is the length of the inlet, and N_{p} (dimensionless) is the number of model particles.

Physically, we can interpret R as the number of real molecules per unit time, per unit length in the out-of-plane direction, that are represented by each model particle. Because this source term acts on the time derivative of the accumulated variable, each particle leaves behind a “trail” in the mesh elements it passes through, which contributes to the number density in those elements.

I have created a second animation in which the number density of molecules is computed using the Accumulator (bottom) and the result is compared to the result of the *Free Molecular Flow* interface (top). Here it is:

We do see some noise in the particle tracing solution because each particle can only make a uniform contribution to the mesh element it is currently in. However, when the number of particles is large compared to the number of mesh elements, it is still possible to obtain an accurate solution.

In addition to the generic Accumulator node, which can represent anything, dedicated accumulator-based features are available in the different physics interfaces, including the following:

- In the
*Charged Particle Tracing*physics interface:- Particle-Field Interaction computes the charge density of particles, which can then be used as a source term to compute the self-potential of a beam of ions or electrons. It is also possible to compute the current density, which can create a significant magnetic field if the beam is relativistic.

- In the
*Particle Tracing for Fluid Flow*physics interface:- Fluid-Particle Interaction computes the body load exerted by particles on the surrounding fluid.

- In the
*Geometrical Optics*physics interface:- Deposited Ray Power generates a heat source term based on the amount of power absorbed by the medium if rays propagate through an absorbing medium.

The third variety of Accumulator is a bit more advanced than the previous two. A *Nonlocal Accumulator* is used to communicate information from a particle’s current position to the initial position from which it was released. The Nonlocal Accumulator can be added to an “Inlet” node, causing it to declare an accumulated variable on the mesh elements on the Inlet boundary.

The Nonlocal Accumulator can be used in some advanced models of surface-to-surface radiation. In many cases, the *Surface-to-Surface Radiation* physics interface (available with the Heat Transfer Module) can be used to efficiently and accurately model radiative heat transfer. However, the *Surface-to-Surface Radiation* interface relies on the assumption that all surfaces reflect radiation diffusely. That is, the direction of reflected radiation is completely independent of the direction of incident radiation. It cannot be used, for example, if some of the radiation undergoes specular reflection at smooth, polished, metallic surfaces.

One approach to modeling radiative heat transfer with a combination of specular and diffuse radiation is to use the *Mathematical Particle Tracing* interface, as demonstrated in the example of mixed diffuse and specular reflection between two parallel plates.

The incident heat flux on each plate is computed by releasing particles from the plate surface, querying the temperature of each surface the particles hit, and communicating this information back to the point at which the particles are initially released. The below image shows the temperature distribution between the two plates, where the top plate is heated by an external Gaussian source.

We have seen that Accumulators can be used to model interactions between particles or rays and any field that is defined on the surrounding domains of boundaries. The accumulated variables can represent any physical quantity. The Accumulator is the basic building block that allows for sophisticated one-way or two-way coupling between a particle- or ray-based physics interface and any of the other products in the COMSOL product suite.

The Accumulators and related physics features have too many settings and applications to discuss in detail in a single blog post. To learn more about the many options available, please refer to the User’s Guide for the Particle Tracing Module (for particle tracing physics interfaces), the Ray Optics Module (for the *Geometrical Optics* interface), or the Acoustics Module (for the *Ray Acoustics* interface).

If you are interested in learning more about any of these products, please contact us.

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Whether on your way to the airport or simply passing by one, you have likely experienced watching a plane fly right overhead as it prepares to land. It is often a sight at which we marvel, to see a plane flying so low to the ground, but another element that can be quite captivating is the sound that the aircraft makes. While our experience lasts only for a short moment, imagine what it is like for those residents living in close proximity to the airport, hearing that sound periodically throughout the day. Taking this perspective, it is easy to see why addressing aircraft noise has become such a prevalent area of concern.

Since becoming a public issue in the 1960s, new regulations and research have initiated the development of quieter aircraft. One design element that has proven successful within this movement is a high-bypass turbofan engine. Used in the majority of airliners, these engines feature a fan that captures incoming air. As the air passes through the fan, a portion enters the combustion chamber, while the remainder bypasses the engine. Compared to its predecessor, the *turbojet*, in which all air passes through the gas generator, the turbofan engine creates less aircraft engine noise while also offering greater thrust at lower speeds.

*A CFM56 turbofan engine. (“CFM56 P1220759” by David Monniaux — Own work. Licensed under Creative Commons Attribution Share-Alike 3.0, via Wikimedia Commons).*

With this improved engine technology, the next step becomes analyzing the acoustical field of the turbofan engine in an effort to optimize its design. For this, we can turn to simulation.

To analyze aircraft engine noise, we can use the Flow Duct model in COMSOL Multiphysics. This model features an axially symmetric duct within a turbofan engine. It is an approximate model of a turbofan’s inlet section in the CFM56 series (which is quite common among airliners). In this example, it is assumed that the flow of air is compressible, irrotational, has no viscosity, and constant entropy. The acoustic field is modeled as perturbations on top of the background flow using the linearized potential flow equations. With these formulations, the pressure and the velocity field are directly related to and derived from a so-called velocity potential.

*The geometry of the duct.*

In this model, *z* = 0 is referred to as the *source plane* and represents the fan’s location in the actual geometry of the engine. A noise source is introduced at this boundary. Meanwhile, *z* = *L* represents the engine’s fore end and is known as the *inlet plane*. Variables R_{1} and R_{2} indicate the spinner and duct-wall profiles.

In this study, we model cases with and without a compressible irrotational background flow. With Mach number *M* = -0.5 (flow in the negative *z*-direction) and *M* = 0 (no flow), respectively. The analysis also compares the use of hard and lined walls within the duct.

The model first solves the background flow, which is assumed to be stationary. Then, a suitable acoustic source (a given propagating eigenmode) is derived. Finally, the acoustic field is found.

For the case with a mean background flow (*M* = -0.5), the velocity potential was found to be uniform beyond the terminal plane (contour lines in the plot below). Additionally, the deviations in the mean density value (due to the compressible nature of the air flow) were most prevalent in nonuniform areas of the duct’s geometry, such as the tip of the spinner. These deviations are highlighted by the red and blue colors in the figure below.

*Plot of mean-flow field for source plane *M* = -0.5. The color surfaces correspond to the background density and the contour plots to the velocity potential.*

Using these results, we can then calculate eigenmodes for the acoustic field at the source of the noise. These can be shown to represent a certain component of the engine noise source at a given frequency. The graph below shows the resulting velocity-potential profile for the first axial boundary mode at the source plane in the case of *M* = -0.5 and *M* = 0.

*Graph showing the resulting velocity-potential profile for the lowest mode.*

With the background flow and a source at hand, the acoustic field can now be solved for. The results (seen below) can be compared to results for a similar system presented by Rienstra and Eversman (2001).

In the cases without a background flow, the acoustic pressure distribution for both hard and lined walls were in good agreement when compared to the simulation results in the paper. With the mean flow, the results from the hard wall case paired well with alternate solutions. However, in the case of the lined wall, there were some notable differences, specifically near the source plane. These differences can be explained by the discrepancy in the noise source definition. In this model, the source mode was derived for the hard duct wall case, while the compared simulation used a noise source adapted to the acoustic lining.

*Pressure distribution in the acoustic field for hard (top) and lined (bottom) duct wall in cases with no mean flow (*M* = 0).*

*Pressure distribution in the acoustic field for hard (top) and lined (bottom) duct wall in cases with a mean flow (*M* = -0.5).*

The model presented here is very conceptual, but it could potentially be extended for more complex situations. By modeling these systems, it is possible to optimize the shape of certain engine duct parts and the lining properties in order to reduce the sound emission. Such optimization should, of course, go hand-in-hand with a control of the flow properties in order to not degrade the engine’s performance.

- Try it yourself: Download the Flow Duct model now.
- S.W. Rienstra and W. Eversman, “A Numerical Comparison Between the Multiple-Scales and Finite-Element Solution for Sound Propagation in Lined Flow Ducts,” J. Fluid Mech., vol. 437, pp. 367–384, 2001.

The researchers at Argonne National Lab (Argonne) turned to multiphysics simulation and trial-and-error prototyping to optimize the effectiveness of their acoustic levitator. When we want to move an object, sound may not be the tool we would typically reach for. So how does it have the power to float or levitate objects in a lab setting? It’s all about combining forces in just the right way to create lift.

When sound vibrations travel through a medium like air, the resulting compression is measurable and real. By combining acoustophoretic force, gravity, and drag, the pressure is just enough to not only lift a material like liquid medicine, but to also allow the medicine to be positioned, rotated, and moved according to the needs of the operator.

*Pressure pockets created by waves between the transducers of the acoustic levitator do the heavy lifting on a particle scale.*

By keeping the droplets in a steady rotation, researchers are able to work on the chemical reactions while the medicine stays liquid and amorphous. This is key for creating a safe, steady environment where medicine will form correctly.

Every material and measurement in the acoustic levitator will change both whether the device works in its final design and how finely it can be adjusted according to the needs of the scientists who use it.

The geometry of the device includes two small piezoelectric transducers that stand like trumpets above and below the working area where medicine is created, like this:

*The acoustic levitator’s wave patterns are controlled by pieces of Gaussian profile foam located on evenly-spaced transducers.*

Possibly the most important part of the design is the Gaussian profile foam, which consists of polystyrene and coats the ends of each transducer. This foam works to remove acoustic waves that fall outside the required range. It acts as a filter to maintain even, well-defined standing waves.

Using COMSOL Multiphysics together with the Acoustics Module, CFD Module, and Particle Tracing Module, the team at Argonne modeled the acoustic levitator. Working cohesively with simulation, they were able to narrow down the shape of the acoustic field and location of floating droplets.

*The simulation above shows that at T = 0.75 seconds droplets formed from the particles. On the left, the simulation shows the expected particle distribution and on the right, a photograph depicts the actual distribution of the droplets.*

As advances in acoustic levitation expand, the ability to work with finer and finer chemical reactions will allow members of the pharmaceutical science community to expand their reach, perhaps discovering many new medicines with life-saving qualities.

- Learn more about how Argonne improved their acoustic levitation technology.

Sound Navigation and Ranging (SoNaR, more commonly written in all lowercase as “sonar”) technology can be used for investigating and communicating underwater. To improve the sonar system, you need to optimize the design at a component level. A major component of sonar is the *electro-acoustic transducer*.

Sonar technology can be used for various applications for the main purpose of detecting objects in water. Some specific applications include mapping the ocean floor for nautical charts, finding hazardous or lost objects, communicating with other vessels, detecting enemy submarines, navigating the seas (both on and underwater), and more.

*Towed sonar. Image credit: http://www.netmarine.net/ via Wikipedia.*

A recent real-world example is the search effort for Malaysia Airlines Flight 370 in April this year. After weeks of employing other search methods, officials decided to use sonar to attempt locating the missing aircraft. The black box of any airplane comes equipped with underwater locator beacons for this very reason. The search team’s sonar did detect signals, but they were unfortunately unable to confirm that they came from Flight 370.

There are two types of sonar: active and passive. Active sonar implies that the sonar device can itself make sounds and then “listen” for the echo to return. The sound signals are created from electrical pulses that are converted into sound via the piezoelectric or magnetostrictive material in the center of a transducer. By transmitting sound and then actively awaiting its return and eventually receiving the signal or echo that bounces back, the device can measure how far away the object in question is.

Passive sonar simply involves listening to sounds made by other objects or beings, such as the locator beacons in the case of Flight 370. Both passive and active sonar systems are able to listen to incoming sounds by converting these into electrical signals, again via the piezoelectric or magnetostrictive material in the transducer.

The sonar performance is only as good as its components. The component responsible for sending and receiving signals is the *electro-acoustic transducer*. For efficiency, you’ll often have a multitude of these transducers arranged in arrays. There are various different designs to choose from, including tonpilz, ring, and flextensional transducers. Here, we will focus on the tonpilz piezo transducer.

A tonpilz piezo transducer contains a stack of active piezoceramic rings in between a light head and heavy tail mass. This assembly allows the transducer to act as either a source or a receiver. Additionally, it could be pre-stressed by a central bolt.

*Tonpilz piezo transducer.*

When designing a tonpilz transducer, you will need to consider several elements. The design is based on multiphysics couplings between acoustics and structural mechanics as well as piezoelectricity and structural mechanics. We want to understand how the device deforms and where stresses lie; what the sound pressure level and radiated pressure field are; and the sound beam’s pattern, transmitting voltage response (TVR) curve, and directivity index (DI).

Due to the multiphysics nature of the component, I would suggest you model it using COMSOL Multiphysics and the Acoustics Module. The Acoustics Module comes with the *Acoustic-Piezoelectric Interaction, Frequency Domain* interface, which contains all the necessary multiphysics couplings for modeling the transducer.

If we open the solved Tonpilz Piezo Transducer model from the Model Gallery, we can study the performance of a transducer with a bolt that is not pre-stressed. Below, you can see four of the most important plots.

Note: My colleague Mads Herring Jensen recently updated the model entry with files for COMSOL Multiphysics 4.4. It is available in the Model Gallery for those of you who want to download the model MPH-file and the accompanying PowerPoint® presentation.

*The sensitivity, or rather the TVR (Transmitting Voltage Response), of the transducer plotted between 1 and 40 kHz.*

*The transducer directivity (depicted as a 3D polar plot) evaluated at a distance of 10 m in front of the transducer for all the modeled frequencies. The normalized directivity is shown in the figure below.
*

*Here, the spatial sensitivity of the transducer is depicted in the* xz*-plane at a distance of 10 m. The patterns are normalized to 0 dB in front. Evaluating this data at any desired distance is a simple postprocessing task. Based on the far-field data, you can also readily calculate the directivity index (DI). This is done in the model.*

*Specific acoustic impedance at the surface of the transducer.*

From plotting the results, we can conclude that the transducer is very versatile. That is because we can control its directivity (varying the direct index from -7 dB to +8 db) given that the TVR is almost constant in the frequency range of 10 to 30 kHz.

- Download the Tonpilz Piezo Transducer model
- Watch how to build the model in the Tonpilz Piezo Transducer Tutorial video
- Learn what you can simulate with the Acoustics Module
- COMSOL Conference papers:

Ever since the first offshore wind farm was built off the Danish coast in 1991, offshore wind has been gaining in popularity. Just over two decades later, at the end of 2012, the European Union was producing enough electricity from offshore wind farms to power approximately five million households. In the coming decade, offshore wind farms are expected to generate nearly one fifth of the European Union’s power consumption, jumping from about 6.04 GW in 2013 to over 150 GW by 2030, according to a report by the European Wind Energy Association.

*Windmill park in Oresund between Copenhagen, Denmark and Malmo, Sweden. Photo credit: Ziad, Wikipedia Commons.*

With this huge increase in wind power expected, engineers are being called in to investigate the effect that offshore turbines could have on marine life. In a recent report conducted by Xi Engineering Consultants for the Scottish Government, Brett Marmo, Iain Roberts, and Mark-Paul Buckingham investigated how different types of wind turbine foundations affect the vibrations that propagate from the turbine into the sea, and ultimately how these vibrations could affect surrounding marine life. Also involved in the project were Ian Davies and Kate Brookes of Marine Scotland, who helped define the water depth, turbine size, and foundation types of the turbines modeled in the study based off of the types of turbines submitted to the Scottish Government for licensing permits. Additionally, Davies and Brookes helped identify the marine species most likely to be affected by offshore wind.

I recently interviewed Brett Marmo about the project. “In our research, we explored how different bases affect the noise that is produced by offshore turbines, and whether or not this noise was loud enough to be heard by marine life,” Marmo explained. “We studied three different wind turbine bases and examined the possible effect that noise could have on various types of local whales, porpoise, seals, dolphins, trout, and salmon.”

Vibrations produced by offshore turbines travel from the tower into the turbine foundation and are released as noise into the surrounding marine environment. “Because the noise is emitted at the interface between the foundation and seawater, it’s likely that the intensity and frequency of the noise will vary with the type of foundation used,” described Marmo. “Using finite element analysis, we modeled three identical wind turbines, only altering the structure of the foundation.”

Below, you can see the three most common foundation types: the gravity base, jack foundation, and monopile foundation. Generally, the jacket and gravity base are used in water 50 meters or deeper, while the monopile is generally not used at depths exceeding 30 meters. Due to the different structures, materials, and size of each of these bases, the vibrations that propagate through the base behave very differently, leading to noise produced with different frequencies and sound pressure levels (SPL).

*Three different foundation types are shown: a gravity base structure sitting on the seabed (left), a jacket with pin pile connections to the seabed (middle), and monopile placed onto the seabed with a transition piece (right).*

“Using simulation allowed us to model the noise produced by the foundations under identical operating conditions — something that we wouldn’t have been able to achieve by just taking measurements of in-service wind turbines,” says Marmo. “Without simulation, the different environments and wind loads that these turbines experience would have made it very difficult to determine if it was truly the foundation that was affecting the noise produced and not another unaccounted for variable.”

Before delving into the simulations, let’s first explore where it is that the noise itself comes from. Noise from wind turbines can come from two places; aerodynamic noise is produced by the blades slicing through the air, and mechanical noise is generated by machinery housed in the gearbox. Almost all of the noise produced by the blades themselves gets reflected back from the water’s surface due to the large refractive difference between the air and water, and does not enter the marine environment.

Therefore, the majority of noise is created by mechanical vibrations produced in the turbine’s gearbox and drivetrain by rotational imbalances, gear meshing, blade pass, and by electromagnetic effects between the poles and stators in the generator. Each of these noise sources produce vibrations with a different frequency, which then transmit down the turbine pole and into the foundation. Here is a table of the different frequencies produced and their origin:

Frequency | |
---|---|

Rotational imbalance of rotor | 0.05 to 0.5 Hz |

Rotational imbalance of high-speed shaft between gearbox and generator | 10 to 50 Hz |

Gear teeth meshing | 8 to 1000 Hz |

Electro-magnetic interactions in the generator | 50 to 2000 Hz |

*Frequency bands likely to contain vibration tones produced in the drive train of wind turbines. Table courtesy of Xi Engineering and adapted from their report*.

Once the vibrations enter the foundation, the amplitude of the noise emitted is affected by the size of the excitation force, the frequency of structural resonance, and the amount of damping in the structure. Additionally, higher wind speeds lead to increased torque acting on the rotor, likely meaning that higher noise is emitted.

“Understanding the effect of damping — the dissipation of vibration energy from a structure — was one of the key analyses conducted in our project,” described Marmo. “In general, steel structures such as the jacket foundation have less damping than those built from granular materials, such as the gravity base, which is made of concrete.” The amount of internal damping taking place within a structure will therefore affect the noise emitted by different structures. In order to determine how these factors affected the noise produced, Marmo and the team turned to simulation with COMSOL Multiphysics.

Noise is produced at the interface between the wind turbine foundation and seawater, where the vibration of the foundation oscillates water molecules to produce a pressure wave that radiates from the foundation as sound. Geometric spreading and absorption reduce the intensity of the sound as it propagates farther from the foundation, with high frequency sound being absorbed more quickly and low frequency sound absorbing slower and therefore propagating further.

Marmo analyzed each of the three foundations at three different wind speeds (5 m/s, 10 m/s, and 15 m/s) and found that typically, the higher the wind speed the louder the noise produced. A comparison of the average sound pressure level at a wind speed of 15 m/s at different frequencies for each of the three foundation types is shown below.

*At frequencies lower than 180 Hz, the monopile produces the largest amount of noise. Of the three foundation types, the monopile continues to produce larger SPL values up to 500 Hz. Around 600 Hz, all three foundation types become comparable in average 30 m SPL with the trend of the jacket foundation rising to become the noisiest at frequencies greater than 700 Hz.*

As the graph shows, the jacket base demonstrates the lowest sound pressure level of the three at low frequencies (around 200 Hz and lower). However, at high frequencies, the jacket produces the highest sound pressure level. The monopile and gravity base exhibit comparable sound pressure levels at lower frequencies, while at higher frequencies the gravity base produces the lowest sound pressure level of the three bases. The images below illustrate the sound pressure level around each of the three foundation types at the frequency at which the foundation produces the loudest noise.

Marmo and the team also created a far-field model that used a Gaussian beam trace model to analyze the distances at which a wind farm containing 16 turbines could be heard. As mentioned above, sound at lower frequencies tends to propagate farther than sound at higher frequencies. Additionally, ambient noise can mask the sound produced by wind turbines, making them nearly impossible to hear. This was also taken into account in Marmo’s analyses.

“We found that each of the different bases produced the loudest sound in the far-field at different frequencies,” described Marmo. “At a wind speed of 10 and 15 m/s, the monopile and gravity bases are audible at least 18 km away at most frequencies below 800 Hz, while the jacket is audible at 250 Hz 10 km away and 630 Hz at least 18 km away.” Here is a summary of these results:

The next step in the project was to determine the frequencies at which marine species could detect the sound and over what distances. Each of the different foundation types emitted different sound pressure levels at different strengths and frequencies. Since various marine animals have different hearing thresholds, this also had to be taken into account.

Cormac Booth and Stephanie King of SMRU Marine at St. Andrews University were the key marine biologists who analyzed the hearing thresholds of different marine species and determined whether or not the noise produced could affect the animal’s behavior.

*Hearing thresholds for dolphins, minke whales, porpoises, and seals.*

Of the species examined, the minke whale had the most sensitive hearing at low frequencies (less than 2000 Hz) and was able to hear the turbine from the farthest distances. “We predict that minke whales will be able to detect wind farms constructed of either monopile or gravity foundations up to 18 km away at most frequencies below 800 Hz and for all three wind speeds,” says Marmo. “On the other hand, bottlenose dolphins and porpoises are less sensitive to low frequencies. Dolphins can detect a wind farm on a gravity base 4 km away at wind speeds above 10 ms, but can only detect jackets and monopiles at close ranges of less than 1 km.”

You can view an example of the results found in Marmo’s report, showing the hearing threshold of a seal for different wind speeds and frequencies:

Determining behavior responses was harder to predict. Using a sensation parameter, Booth and King estimated the upper and lower ranges around the hearing threshold of each of the species. Then, they determined what percentage of animals could be expected to move away from the turbines within a certain sound pressure range.

Neither seal species nor bottlenose dolphins were predicted to exhibit a behavioral response to the sounds generated under any of the operational wind turbine scenarios. However, between around 4 kilometers and 13 kilometers, 10 percent of minke whales encountering the noise field produced by the monopile foundation were expected to move away. Overall, jacket foundations appear to generate the lowest marine mammal impact ranges when compared to gravity and monopile foundations.

What does this mean for the future of offshore wind power? Marmo and his team’s report found that there were little to no detrimental effects from wind turbine noise on marine species. Although more studies still need to be conducted, these findings demonstrate that the future of offshore wind is looking positive.

- Explore the full report by Xi Engineering “Modelling of Noise Effects of Operational Offshore Wind Turbines including noise transmission through various foundation types“
- “Offshore Wind May Provide One-Fifth of EU Electricity“
- Check out this resource: Cape Wind, a proposed farm that will likely become the first offshore wind farm in America

When modeling acoustics phenomena using the *Thermoacoustics* interface, there are several things to be aware of. First off, the physics have to be set up correctly and the mesh has to resolve the viscous and thermal boundary layers. It is also important to note that solving a thermoacoustic model involves solving for the pressure, velocity field (for example, 3 components in 3D), and temperature. This means that the model can become computationally expensive and involve many degrees of freedom (DOFs).

Erroneous specifications of the coefficients of thermal expansion and compressibility is a problem that I often see in support cases. If these coefficients are wrong or even evaluate to zero, the result is a model where acoustic waves (pressure or compressibility waves) propagate at the wrong speed of sound or do not propagate at all. The speed of sound relates to both of these coefficients.

A detailed description of both of these coefficients and on how to define them is given in the Acoustics Module’s *User’s Guide* (under The *Thermoacoustics, Frequency Domain* Interface in the section *Thermoacoustics Model*). The model Vibrating Particle in Water: Correct Thermoacoustic Material Parameters, which can be found in the Model Library, also discusses these issues. A simple check is to plot the parameters `ta.betaT`

(isothermal compressibility) and `ta.alpha0`

(thermal expansion) after solving the model to ensure that they have the correct values.

When meshing a thermoacoustics model, it is important to properly resolve the acoustic boundary layer to capture the physics correctly. In order to do this and avoid too many mesh elements, there are a few tricks you can use:

*Create parameters to control your mesh.*For example, create a parameter for the analysis frequency, say`f0`

, and then also create a parameter for the viscous (or thermal) boundary layer thickness at this frequency. In air, we know that the viscous boundary layer thickness at 100 Hz is 0.22 mm, and, in general, you can write the thickness as`dvisc = 0.22[mm]*sqrt(100[Hz]/f0)`

. If you perform a frequency sweep, you can create parameters for the thickest and thinnest value of the boundary layer. Having these parameters at hand can help you build a good mesh.*Use Boundary Layers.*This will keep the number of mesh elements constant for all studied frequencies. This is especially important in 3D. If you simply prescribe a maximum element size on the walls, the number of mesh elements will explode as the boundary layer thickness decreases.*Use logic expressions when defining the mesh.*For example, use`min(,)`

when defining the maximum element size or the thickness of a boundary layer. In the figure below, an example is given of a circular duct with a diameter 2a = 2 mm. The overall “Maximum element size” is set to a/3. A boundary layer mesh is used with five layers and a thickness of`min(a/30,0.3*dvisc)`

. This ensures a constant mesh thickness up to around 500 Hz (keeping the mesh in the middle of the pipe of good quality) and then the thickness decreases with`dvisc`

as the frequency parameter`f0`

increases.

In general, when solving a model using the Frequency Domain study step, it is not possible to have the mesh depend on the frequency variable `freq`

. This is what you would like for this type of models. However, it is possible to achieve this when performing a parametric sweep. Therefore, one workaround is to use a Parametric Sweep around the Frequency Domain study step. Sweep the parameter over `f0`

and set `f0`

to be the frequency in the Frequency Domain step.

Note that when doing this, the COMSOL software will re-mesh every time a parameter in the mesh changes, which may slow down the computation a bit. On the other hand, you can set up a more intelligent mesh in this way and still save time.

A final option is to prepare several meshes, maybe one mesh for each chunk of 1000 Hz, and then use several studies with these meshes selected for a restricted frequency range.

*Example of a mesh that captures the effects in the acoustic boundary layer, here shown at four different frequencies. The color represents the RMS velocity for a wave traveling in an infinite circular duct with a diameter of 2 mm.*

In that it is computationally expensive to solve thermoacoustics models, it is often advantageous to do so only in the parts of your system where thermoacoustics is relevant. These simulations can then be combined with simulations based on less-complex physics that describe the rest of your system. Here are some ideas on how this can be done:

*Couple the thermoacoustics model to pressure acoustics where relevant.*In models where large differences exist in the geometry scale, only use thermoacoustics in the narrow regions and pressure acoustics in the larger domains. The*Thermoacoustics*interface is a multiphysics interface that has the ability to be automatically coupled to the*Pressure Acoustics*interface. This is exemplified in the Generic 711 Coupler model (located in both the Model Library within the software and the Model Gallery on our website).*Use submodels and lumped models.*For instance, extract a transfer impedance from a detailed thermoacoustic model and use it in a pressure acoustics model. A nice example model of this is seen in the Acoustic Muffler with Thermoacoustic Impedance Lumping model. In this example, the transfer impedance of a perforated plate is analyzed and used in a pressure acoustics model.*As frequency increases, the acoustic boundary layer decreases in size and relevance.*This means that at a certain frequency, the boundary layer losses can be considered to become negligible, and you can switch to solving the modeling as a pressure acoustics problem.- In structures of constant cross section you can use the Narrow Region Acoustics models of the
*Pressure Acoustics*interface. These are homogenized fluid models where the boundary layer losses are smeared over the fluid domain. These models provide a first good approximate response of a system without the cost of solving a full thermoacoustic model.

The documentation for the *Thermoacoustics* interface contains some tips and tricks on how to use different solver approaches if the model becomes very large. See: Acoustics Module User’s Guide > The Thermoacoustics Branch > Theory Background for the Thermoacoustics Branch > Solver Suggestions for Large Thermoacoustic Models.

The most important points when modeling acoustics using the *Thermoacoustics* interface are:

- Solve only for thermoacoustics where and when necessary; investigate if the viscous and/or thermal boundary layer thickness are comparable to the geometrical scale or not (depending on the frequency range and geometry scales).
- Check material parameters to be sure that both compressibility and thermal expansion are non-zero.
- Check the mesh size at boundaries and compare it to the viscous and thermal boundary layer thickness.

Examples of systems where the use of thermoacoustics is important are listed below.

Electroacoustic transducers are a good example of true multiphysics models where it is essential to include both thermal and viscous losses:

- Blog post: Thermoacoustics Simulation for More Robust Microphone Analysis
- Model downloads:
- B&K 4134 Condenser Microphone, results compared with measurements
- Tutorial model on a simplified 2D axisymmetric condenser microphone model

- COMSOL News article about the use of COMSOL to model hearing aids, “Simulation-Based Design of New Implantable Hearing Device“

- Blog post about using COMSOL Multiphysics to model MEMS microphones

- An example of a vibrating micromirror, which solves for thermoacoustics in order to model Fluid Structure Interaction (FSI) in the frequency domain

The solution of a Thermoacoustics sub-model to find the transfer impedance of a perforated plate in a muffler system. The impedance is subsequently used as a transfer impedance condition in a Pressure Acoustics model:

- Model download: Acoustic Muffler with Thermoacoustic Impedance Lumping

Modeling the response of an Ear Canal Simulator, the so-called 711 coupler. The model results are compared to IEC standard curves and to a lossless model. The results clearly show the necessity to include thermal and viscous losses.

- Model download: Generic 711 Coupler an Occluded Ear Canal Simulator

Advanced application using the *Thermoacoustic* interface to model photoacoustic applications.

- Model download: Photoacoustic Resonator

- A thermoacoustic tutorial model describing the importance of setting up the compressibility and thermal expansion material parameters correctly

- COMSOL Documentation: Acoustics Module User’s Guide.
- COMSOL Documentation: Acoustics Module User’s Guide > The Thermoacoustics Branch.

It is, for example, necessary to include the thermal and viscous losses when modeling the response of small transducers, like condenser microphones, MEMS microphones, and miniature loudspeakers (i.e. receivers). Other applications include analyzing feedback in hearing aids and in mobile devices, or studying the damped vibrations of MEMS structures.

A good example for us to investigate here, which relates to an engineering application, is the transfer impedance of the standard IEC 60318-4 occluded ear canal simulator (sometimes referred to as the 711-coupler), as depicted in the figure below. In the graph to the right, the response is modeled including and excluding thermoacoustic losses. It is evident that these types of losses need to be included in order to capture the correct behavior when comparing their curves to the standard simulator’s data.

*The pressure distribution inside an occluded ear canal simulator at 7850 Hz, complying with the IEC 60318-4 standard, is depicted to the left. The modeled transfer impedance of the coupler (in blue, including thermal and viscous losses) is shown together with the prescribed standard curves (in red), and the curve resulting from a pure lossless model (in green).*

The thermoacoustic effect is typically seen and is most pronounced at resonances, which are rounded and shift down in frequency. To model these effects, it is necessary to include thermal conduction effects and viscous losses explicitly in the governing equations, solving the momentum (Navier-Stokes), mass (continuity), and energy conservation equations. This is achieved by solving the thermoacoustics equations in the *Thermoacoustic* interface, included in the Acoustics Module. The equations are also known as the thermo-viscous acoustics, visco-thermal acoustics, and linearized Navier-Stokes equations.

Here, we will present the physical background for the thermoacoustics equations along with the important boundary layer characteristic, length scale. We will also provide a short description of the material parameters necessary for describing fluid media.

Acoustic waves are the propagation of small linear fluctuations in pressure on top of a background stationary (atmospheric) pressure. The governing equations for the fluctuations (the wave equation or Helmholtz’s equation) are derived by perturbing, or *linearizing*, the fundamental governing equations of fluid mechanics — the Navier-Stokes equation, the continuity equation, and the energy equation. Doing this results in the conservation equations for momentum, mass, and energy for any small (acoustic) perturbation.

For many applications simulating acoustics, a series of assumptions are then made to simplify these equations: the system is assumed lossless and isentropic (adiabatic and reversible). Yet, if you retain both the viscous and heat conduction effects, you will end up with the equations for thermoacoustics that solve for the acoustic perturbations in pressure, velocity, and temperature.

The procedure to derive the governing equations in the frequency domain is to assume small harmonic oscillations about the steady background properties. The dependent variables take the form:

p = p_0+p’e^{i\omega t}, \quad \mathbf{u} = \mathbf{u}_0+\mathbf{u}’ e^{i\omega t}, \quad T = T_0 + T’ e^{i\omega t}

where p is the pressure, \mathbf{u} is the velocity field, T is the temperature, and \omega is the angular frequency. Primed (‘) variables are the acoustic variables, while variables accompanied with the subscript 0 represent the background mean flow.

In thermoacoustics, the background fluid is assumed to be quiescent so that \mathbf{u}_0=\mathbf{0}. The background pressure p_0 and background temperature T_0 need to be specified (they can be functions of space). Inserting the above equation into the governing equations and only retaining terms linear in the first-order variables yields the governing equations for the propagation of acoustic waves including viscous and thermal losses.

Note: Details on this can be found in the User’s Guide of the Acoustics Module in the “Theory Background for the Thermoacoustic Branch” section.

The governing equations in the *Thermoacoustic* interface, in the frequency domain, are the continuity equation (omitting primes from the acoustic variables):

i\omega\rho =-\rho_0 (\nabla\cdot\mathbf{u})

where \rho_0 is the background density; the momentum equation:

i\omega\rho_0 \mathbf{u} = \nabla\cdot \left(-p\mathbf{I}+\mu ( \nabla\mathbf{u}+(\nabla\mathbf{u})^T )+\left(\mu_\textrm{B}-\frac{2}{3}\mu \right)(\nabla\cdot\mathbf{u})\mathbf{I} \right)

where \mu is the dynamic viscosity and \mu_\textrm{B} is the bulk viscosity, and the term on the right hand side represents the divergence of the stress tensor; the energy conservation equation:

i\omega (\rho_0 C_p T -\ T_0 \alpha_0 p) = -\nabla\cdot(-\textrm{k}\nabla T) + Q

where C_p is the heat capacity at constant pressure, \textrm{k} is the thermal conductivity, \alpha_0 is the coefficient of thermal expansion (isobaric), and Q is a possible heat source; and finally, the linearized equation of state relating variations in pressure, temperature, and density:

\rho = \rho_0 (\beta_T p -\alpha_0 T)

where \beta_T is the isothermal compressibility.

The left-hand sides of the governing equations represent the conserved quantities: mass, momentum, and energy (actually entropy). In the frequency domain, multiplication with i\omega corresponds to differentiation with respect to time. The terms on the right-hand sides represent the processes that locally change or modify the respective conserved quantity. In two of the equations, diffusive loss terms are present, due to viscous shear and thermal conduction. Viscous losses are present when there are gradients in the velocity field, while thermal losses are present when there are gradients in the temperature.

When sound waves propagate in a fluid bounded by walls, so-called *viscous* and *thermal boundary layers* are created at the solid surfaces. At the wall, the no-slip condition applies to the velocity field, \mathbf{u} = 0, and an isothermal condition for the temperature, namely T = 0. The isothermal condition is a very good approximation, as thermal conduction is typically orders of magnitude higher in solids than fluids. These two conditions give rise to the *acoustic boundary layer*, which consists of the viscous and a thermal boundary layers. The flow transforms from the bulk condition of being nearly lossless and described by isentropic (adiabatic) conditions to the conditions in this layer.

The problem of a time-harmonic wave propagating in the horizontal plane along a wall (this could be waves propagating in a small section of a pipe) is illustrated in the figures below. The left figure shows the velocity amplitude and the right figure the fluid’s temperature, from the wall towards the bulk, while the middle figure shows the velocity magnitude as well as an animation indicating the velocity vector over a harmonic period.

*Velocity amplitude (left) and fluid temperature (right), from the wall to the bulk, of an acoustics wave propagating in the horizontal plane (bottom). The viscous and thermal boundary layer thicknesses are indicated by the red dotted lines closest to the wall. The upper dotted lines represent 2 \pi times the boundary layer thickness, in each case. The animation indicates the acoustic velocity components, while the color plot shows velocity amplitude.*

The viscous and thermal boundary layers are clearly visible. Because gradients are large in the boundary layer, losses are large here too. This means that in systems of relatively small dimensions, the losses associated with the boundary layer become important. In many engineering applications (miniature transducers, mobile devices, etc.), including the losses associated with the boundary layer is essential in order to model the correct physical behavior and response.

The viscous characteristic length is shown as a red dotted line in the velocity and temperature plots shown above, together with 2 \pi times the value (known as the viscous/thermal wavelength). The two characteristic lengths are related by the dimensionless Prandtl number Pr:

\textrm{Pr} = \frac{C_p \mu}{\textrm{k}} \qquad \delta_\textrm{visc} = \sqrt{\textrm{Pr}} \: \delta_\textrm{therm}

which gives a measure of the ratio of the viscous to thermal losses in a system. For air, this number is 0.7, while it is around 7.1 for water. In air, the thermal and viscous effects are roughly equal in importance, while for water (and most other fluids), the thermal losses only play a more minor role. The viscous and thermal boundary layer thicknesses exist as pre-defined variables for use in postprocessing in the Acoustics Module, and they are denoted by `ta.d_visc`

and `ta.d_therm`

. The Prandtl number is denoted by `ta.Pr`

.

The plane wave problem can be solved analytically and expressions for the viscous (d_\textrm{visc}) and thermal (d_\textrm{therm}) boundary layer thickness subsequently derived. They are given by:

\delta_\textrm{visc} = \sqrt{\frac{2\mu}{\omega\rho_0}} \qquad \delta_\textrm{therm} = \sqrt{\frac{2 \textrm{k}}{\omega\rho_0 C_p}}

The value of d_\textrm{visc} is 0.22 mm for air and 0.057 mm for water at 100 Hz, 20°C and 1 atm. Over a range of frequencies, the viscous and thermal boundary layer thickness can be plotted, such as the figures below:

*The value of the viscous (d_\textrm{visc}) and thermal (d_\textrm{therm}) boundary layer thickness as functions of frequency for (left) air and (right) water.*

This shows the diminishing effect of viscous and thermal losses at increasing acoustic wave propagation frequencies. Finally, another important effect that is captured when modeling with the Thermoacoustic interface is the transition from adiabatic to isothermal acoustics at low frequencies in small devices. This effect occurs when the thermal boundary layer stretches over the full device and is important in, for example, condenser microphones, such as the B&K 4133 condenser microphone. At isothermal conditions the speed of sound changes to the isothermal speed of sound.

It is important to note that viscous and thermal losses also exist in the bulk of the fluid. These are losses that typically occur when acoustic signals propagate over long distances and are attenuated. One example of this is sonar signals. These types of losses are, in air, only dominating at very high frequencies (they can be neglected at audio frequencies). The bulk losses are, of course, also described by the governing equations for thermoacoustics as they include all the physics. However, modeling large domains with the thermoacoustics equations is very computationally expensive. In the Acoustics Module, you should instead use the Pressure Acoustics interface and select one of the available fluid models: *Viscous*, *Thermally conducting*, or *Thermally conducting and viscous*.

Solving a full thermoacoustic model involves defining several material parameters:

- Dynamic viscosity \mu:
- The dynamic viscosity measures the fluid’s resistance to shearing in the fluid. It is the constant that relates stress to velocity gradients. The dynamic viscosity is related to the kinematic viscosity \nu by the relation \mu = \rho_0 \: \nu. The symbol for the dynamic viscosity \eta is also sometimes used.

- Bulk viscosity \mu_\textrm{B}:
- The bulk viscosity is also known as the volume viscosity, the second viscosity, or the expansive viscosity. It is related to losses that appear due to the compression and expansion of the fluid. \mu_\textrm{B} appears in the stress tensor term (right side of equation 3), which has to do with the compressibility (\nabla\cdot\mathbf{u}) of the bulk fluid. This factor is difficult to measure and is often seen to depend on the frequency.

- Heat capacity at constant pressure (specific) C_p:
- This material parameter gives a measure of how much energy is required to change the temperature of the fluid (at constant pressure).

- Coefficient of thermal conduction \textrm{k}:
- The coefficient of proportionality between the temperature gradient and the heat flux in Fourier’s heat conduction law.

- Coefficient of thermal expansion (isobaric) \alpha_0:
- This is the volumetric thermal expansion of the fluid and expresses the ability of the fluid to expand when its temperature rises.

- Isothermal compressibility \beta_T:
- Important parameter in the equation of state of the fluid. It relates changes in pressure to changes in volume in the fluid. The isothermal compressibility is related to the usual (isentropic) compressibility through the ratio of specific heats by \beta_T = \gamma \beta_s.

Now that you know the theory behind thermoacoustics and the associated equations, we can move on to tips and tricks for setting up a thermoacoustic model using COMSOL Multiphysics and the Acoustics Module. We will discuss that as well as examples and applications in the next blog post of this series.

- COMSOL Documentation: Acoustics Module User’s Guide
- COMSOL Documentation: Acoustics Module User’s Guide > The Thermoacoustics Branch
- D. T. Blackstock, “Fundamentals of Physical Acoustics”, John Wiley and Sons, 2000
- S. Temkin, “Elements of Acoustics”, Acoustical Society of America, 2001
- B. Lautrup, “Physics of Continuous Matter”, Second Edition, CRC Press, 2011
- P. M. Morse and K. U. Ingard, “Theoretical Acoustics” Princeton University Press
- A. D. Pierce, “Acoustics; An Introduction to Its Physical Principles and Applications”, Acoustical Society of America, 1989
- A. S. Dukhin and P. J. Goetz, “Bulk viscosity and compressibility measurements using acoustic spectroscopy”, J. Chem. Phys. 130, 124519 (2009)