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)

*The Knowles SPU0409LE5FH MEMS condenser microphone with dimensions 3.76 x 3 x 1.1 mm ^{3}. Photo courtesy of Knowles Electronics*.

A MEMS microphone is a condenser microphone that comprises a MEMS die and a complementary metal-oxide-semiconductor (CMOS) die combined in an acoustic housing. The CMOS often includes both a preamplifier as well as an analog-to-digital (AD) converter. Because of this and the small size of the microphone, it is well suited for integration in digital mobile devices, smart phones, headsets, and hearing aids. The housing with the acoustic port is depicted in the image above. The condenser or variable capacitor consists of a highly compliant diaphragm in close proximity to a perforated, rigid backplate. The perforations permit the air between the diaphragm and backplate to escape. The diaphragm and backplate pair is referred to as the motor (shown in the figure farther down below). The microphone works by first polarizing (charging) the condenser with a DC voltage. This voltage will also result in a static deformation and tensioning of the diaphragm, and, to a much less extent, the backplate. When an acoustic signal reaches the diaphragm through the acoustic port, the diaphragm is set in motion. This mechanical deformation in turn results in an AC voltage across the microphone. These effects combine to provide a real multiphysics problem well suited for analysis in COMSOL Multiphysics. The sensitivity of a microphone is expressed as the ratio of the incident pressure to the measured voltage on the dB scale.

The MEMS microphone model includes a description of the electrical, mechanical, and acoustical properties of the transducer. The acoustic description includes thermal and viscous losses explicitly solving the linearized continuity, Navier-Stokes, and energy equations, that is, *thermoacoustics*. The mechanics of the diaphragm were also modeled including electrostatic attraction forces and acoustic loads, or *electromechanics*. A submodel was also implemented to analyze the interplay between the vibrating diaphragm and the small perforations in the microphone backplate. The model had no free-fitting parameters and it resulted in the prediction of the static mechanical behavior of the MEMS motor (the diaphragm and backplate system) as well as the dynamic frequency response. The model results showed good agreement with measured data.

Because the geometrical dimensions are so small in this system, the vibrations of the diaphragm will be highly damped by the air. The air and acoustics need to be treated including both thermal conduction and viscous losses. The viscous penetration depth (thickness of the acoustic viscous boundary layer) is, for example, 55 µm at 100 Hz and 5.5 µm at 10 kHz, which is larger than or comparable to the distance between the backplate and diaphragm, which is only 4 µm. The *Thermoacoustics* interface of the Acoustics Module is the natural first choice for modeling these effects. This interface will also result in the correct modeling of the transition from adiabatic to isothermal behavior at low frequencies. The complex combined mechanics and electrostatics effects are all included in the *Electromechanics* interface of the MEMS Module. The two physics are fully coupled at the fluid-structure boundary by requiring continuity in the displacement/velocity field.

As a MEMS microphone makes out a complex system, we faced several challenges when trying to model it in detail. Some of these included:

- During the clean room MEMS fabrication process, the diaphragm is released and will bend slightly
- It is important to give a correct description of its initial shape and stress distribution

- The geometry of the microphone is complex, involves many different aspect ratios, and small length-scales
- Thinking about the mesh is important

- Because the system is complex and involves many different physics, the resulting model can easily become too large to solve
- Reducing the model by using symmetries and lumped approximations also needs to be addressed

A classical condenser microphone, like the B&K 4134 from the Model Gallery, in essence works the same way as the MEMS microphone and involves solving the same physics. Modeling it, however, involves some specific challenges as mentioned above. They are primarily due to the complex fabrication involved and lie in describing the initial static state and the complexity of the geometry.

*Sketch of the MEMS microphone motor (not to scale). The diaphragm has a thickness of 1 µm, the gap between the backplate and the diaphragm is 4 µm, the diameter of the perforations in the backplate is 10 µm, and the thickness of the backplate is 2 µm. The distance across the motor from support post to support post is 590 µm. Sketch courtesy of Knowles Electronics*.

As a first step when planning the modeling process, we decided to focus on validating the initial stationary description of the model. One direct measurement of the stationary shape of the microphone is achieved by measuring the DC capacitance as a function of the polarization voltage. The measurements are compared to the model results in the figure below. As you can see, the two curves show good agreement. At about 15.8 V, the measured curve is seen to jump. This corresponds with the point where the diaphragm bends so much due to the electrostatic forces that it touches the backplate.

*Simulation results of the microphone static capacitance as a function of the DC polarization voltage. The green curve represents measurements and the blue curve the modeled capacitance including a constant offset accounting for the constant parasitic capacitance present when performing measurements (0.23 pF). Measurements courtesy of Knowles Electronics*.

The electric potential in slices through a 30 degree cut-out of the microphone motor is depicted in the image below. The field is seen to have very strong gradients in the region where the electrodes are located, while it drops off outside of this region. The holes in the backplate are clearly seen to influence the field. The full dynamic behavior of the microphone was also analyzed solving for the structural displacement, the electric field, and the thermoacoustic fields (pressure, velocity, and temperature) in the frequency domain. This is a fully coupled multiphysics model in a complex geometry and therefore required up to 60 GB of RAM to solve. The resulting sensitivity also showed good agreement with measurements.

*Stationary electric potential depicted in slices through the MEMS microphone motor.*

A unit cell of the diaphragm and backplate system is shown in the below animations. The model represents one hole and the air gap (thin air film) between the vibrating diaphragm and the backplate (here fixed). This system is analyzed as a unit cell using symmetries. The detailed coupled acoustic behavior including viscous and thermal losses is again captured using the *thermoacoustics* interface. The two animations show the behavior of the instantaneous acoustic velocity distribution and temperature distribution at 10 kHz over one period. The model also solves for the pressure (not shown here). The build-up and decay of both the thermal and viscous boundary layers can be seen in the animations. At 10 kHz, the thickness of both is about 5 µm and is comparable to the air gap height of 4 µm.

*Dynamic analysis of one “unit cell” of the diaphragm and backplate system, here modeled at 10 kHz. Shown here are the instantaneous acoustic velocity magnitude (color) and velocity field (vectors).*

*Dynamic analysis of one “unit cell” of the diaphragm and backplate system, here modeled at 10 kHz. The acoustic temperature variations are depicted in this animation.*

- M. J. H. Jensen, W. Conklin, and J. Schultz,
*Characterization of a microelectromechanical microphone using the finite element method*, J. Acoust. Soc. Am.,**134**, pp. 4122 (2013) (Conference abstract) - B&K 4134 Condenser Microphone model in the Model Gallery
- Knowles Electronics
- Acoustics Module
- MEMS Module
- P. Loeppert and S. Lee,
*Sisonic–the first commercialized MEMS microphone*, Solid-state sensors, actuators and microsystems workshop, Hilton Head Island, South Carolina, pp. 27–30 (2006)

Brüel and Kjær is a leading supplier of sound and vibration measurement solutions for various industries around the globe. Part of their vision is to deliver “innovative technical solutions [that] create sustainable value” for their customers. Brüel & Kjær prides itself on being the industry standard in the area of measurement-grade microphones, providing accurate and detailed acoustic measurements. Maintaining the output quality of their products includes the use of COMSOL Multiphysics to refine and improve their microphone analysis devices. Recently, they created a model of a 4134 condenser microphone to simulate the results of different parameters.

*Picture of the Brüel and Kjær 4134 microphone including the
protection grid mounted on the housing. Image courtesy of
Brüel and Kjær Sound & Vibration Measurement A/S.*

Many modeling studies have been performed on the Brüel and Kjær 4134 microphone that did not include all the physical effects acting on the microphone. In the COMSOL Multiphysics simulation however, four different physics interfaces were used to provide a more accurate model. The *Thermoacoustics* interface solves the acoustics in a detailed way by including thermal conduction and viscous losses (this is also known as viscothermal or thermoviscous acoustics, and it solves the linearized continuity, Navier-Stokes, and energy equations); the *Electrostatics* interface measures the change in the electric field and electrostatic forces across the microphone; the *Membrane* interface sets up a pre-tensioned physics for the diaphragm inside the casing; and the *Moving Mesh* interface models the deformation of the diaphragm membrane when pre-polarizing the microphone.

The microphone is essentially an electromechanical transducer that transforms the movements of the diaphragm caused by sound pressure into an electric signal measured in volts. The relationship between the (acoustic) input pressure field P_{in} and the voltage output V_{out} is the sensitivity level L, defined as:

L = 20\log\biggl[\biggl\lvert\frac{P_{\textnormal{in}}}{V_{\textnormal{out}}}\biggl\rvert/\biggl(1\frac{\textnormal{V}}{\textnormal{Pa}}\biggl)\biggl]-L_0

The sensitivity plot (shown below) compares two differing model results with the lower, upper, and average measurement values. As you can see, the results of the model, which have no free tuning parameters, compare favorably with the results from the experimental measurements. The sensitivity of the microphone is an important specification when deciding between varying microphone models. More results, including the membrane deformation, velocity, sound pressure level, temperature, and electric potential, can be found in the model documentation, along with complete instructions on setting up and running the model.

*Microphone sensitivity curve from the model with exposed (blue) and unexposed (pink) vent. Three measurement
curves are also added (green, red, and cyan) to illustrate the variability in the microphone sensitivity.*

Thermal conduction and viscous losses significantly affect acoustic devices, and should be included in all simulations with small-scale geometries. Due to the complicated nature of this physics coupling, all the necessary equations are automatically coupled and solved simultaneously by the software to make it easier for the user. To learn more about thermoacoustics and how to model small acoustics devices, attend the upcoming webinar on “Acoustics Simulation in Microphones, Mobile Devices and Hearing Aids“. The Brüel and Kjær 4134 Condenser Microphone model will be shown to display the different physics described in this post and to demonstrate how they can be included in your own simulations.

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Magnetostriction is an effect that causes a change in shape in all magnetic materials when they are exposed to a magnetic field. For example, a piece of iron would elongate by 0.002% and nickel would contract by 0.007%. This observation generated significant interest during World War I due to the application of the phenomenon in SONAR design. Further research resulted in the creation of engineered magnetostrictive materials such as Terfenol-D and, more recently, Galfenol, which exhibit elongation as large as 0.04 to 0.2%.

The phenomenon of magnetic field induced strain is also known as the *Direct* (magnetostrictive) effect. The magnetostrictive effect can be traced down to the atomic level interactions that arise as a result of the balancing act played by the magnetic and mechanical energy in magnetic materials when they are subjected to a magnetic field and mechanical stress. The animation below is a simple illustration of what goes on inside a magnetostrictive material.

When you cycle the magnetic field applied to the material, it makes the tiny ellipsoidal magnets that make up the material flip back and forth along with the change in the magnitude and direction of the magnetic field. The reorientation of these magnetic building blocks manifests as a macroscopic strain. If you cycle the magnetic field at a typical power line frequency (50Hz – 60Hz), the cyclic strain in the material will make it work like a speaker, thereby producing audible sound. This explains the mystery of the humming sound coming from a transformer.

As a consequence of this bidirectional magnetomechanical coupling, we also see an *Inverse* effect where stress acting on a magnetic material can change the magnetic state of the material itself by reorienting these tiny magnets. The Direct and Inverse effects are respectively used in actuation and sensing applications.

Magnetostrictive materials have something to offer to almost every industry, ranging from aerospace and oil production, to acoustics and MEMS. Some of the important commercial applications are listed below:

- Acoustic devices
- SONAR
- Hydrophone
- Ultrasonic shaker for cleaning, mixing, and emulsification
- Ultrasonic friction welding
- Actuators
- Linear and rotary motors
- Inchworm actuator
- Position controller for machine tool head
- Fuel injection system
- Optical scanning system
- Hydraulic actuators such as servo valves and pumps
- Active trailing edge in smart wings to reduce drag
- Sensors
- Position sensor
- Non-contact torque sensor
- Magnetic field sensor
- MEMS bio and chemical sensors
- Vibration control
- Vibration damper
- Platform stabilizer
- Image stabilizer
- Energy harvester
- Hybrid smart structures
- Tonpilz transducer with hybrid piezoelectric/magnetostrictive core
- Hybrid piezoelectric/magnetostrictive composite actuators and sensors

You can also use them to turn the walls or windows in your living room into speakers!

So, how can you model this interesting phenomenon in COMSOL Multiphysics?

The right recipe for modeling magnetostrictive transducers involves accurately simulating the magnetic and structural behavior, while also capturing the interaction between these physics using an appropriate material model. COMSOL provides you with predefined physics interfaces to set up the magnetic and structural simulations. It also provides you with the flexibility to set up user-defined custom constitutive laws to mathematically represent the material model.

Experimental evidence shows that both the Direct and Inverse magnetostrictive effects are nonlinear. It could be important to model the full nonlinear response when simulating devices that operate under quasi-static conditions but are exposed to a large range of mechanical forces and magnetic fields. In such devices, it may be useful to know under what operating conditions the magnetostrictive core saturates. Such information can provide a designer’s limit and also explain the realistic nonlinear behavior, such as a change in sensitivity of a sensor or the maximum force from an actuator, which a user can expect to get from the magnetostrictive device.

In acoustic transducers that operate at certain known frequencies and under a known set of operating conditions, it is possible to simplify the material model by using linear constitutive laws. These laws (or equations) are derived under the assumption that the transducer operation involves small oscillations about a bias point. Making such practical considerations in the modeling approach allows us to easily simulate the response of magnetostrictive transducers over a wide range of operating frequencies.

In COMSOL Multiphysics, it’s possible to set up both the nonlinear and linearized constitutive equations for modeling magnetostrictive devices. Here, I would like to share with you some snapshots of the results that we got from simulating an experimental transducer.

A typical transducer has a magnetostrictive core surrounded by a drive coil. A magnetic field is produced by the current flowing through the coil. The transducer has a steel housing that encloses the drive coil and the core. The core is attached to a piston that is used to transfer the displacement of the core to an external mechanical component in an actuator configuration, or transfer the load from an external mechanical or acoustic source onto the core in a sensor configuration. The steel housing, piston, and core also create a closed magnetic flux path.

For the nonlinear model, we used typical material characterization curves of Galfenol and were able to identify the nonlinearity in important design parameters, such as the transducer’s blocked force. We were also able to explore the variation in actuation and sensing behavior as a function of a wide range of magnetic fields, and both tensile and compressive loads acting on the transducer. For more information on this model, check out the *Nonlinear Magnetostrictive Actuator and Sensor* tutorial in our Model Gallery.

*Displacement amplitude, actuator and sensor curves, and transducer blocked force plots in a nonlinear magnetostrictive actuator simulation.*

For the linear model, we used typical material parameters of Terfenol-D and were able to generate the actuator load lines. We were also able to explore the frequency response of the amplitude and phase of transducer displacement and drive coil impedance. For more information on this model, I suggest you check out the *Linear Magnetostrictive Transducer* tutorial in the Model Gallery.

*Actuator load line, coil impedance, displacement amplitude, and displacement phase plots in a linear magnetostrictive transducer simulation.*

Not long ago, my colleague Bernt mentioned in a blog post that Dr. Julie Slaughter from ETREMA Products will be delivering a keynote talk on modeling magnetostrictive transducers using COMSOL Multiphysics at the COMSOL Conference 2013 in Boston. If you’re a magnetostriction enthusiast like I am, you would definitely not want to miss the opportunity to sit in on her presentation.

- Download the
*Linear Magnetostrictive Transducer*tutorial from the Model Gallery - Dowload the
*Nonlinear Magnetostrictive Actuator and Sensor*tutorial from the Model Gallery - Understanding Transformer Noise is a great resource explaining the magnetostrictive effect
- Read aboutMains hum

In order to determine the sound produced by an object, you must know all of the structural qualities of that object. Conversely, in order to understand the structural behavior of an object, you must take into consideration the properties of the fluid (such as air or water) surrounding the object and the sound waves produced in that fluid. The *load* (or pressure) of the air on the structure will also affect the ability of the structure to vibrate. That’s where COMSOL comes in, making such a coupling between the structural mechanics and acoustic aspects of an object easy to simulate and analyze.

One application where this acoustic-structural interaction analysis is applicable in music is in the design of a *tuning fork*. A tuning fork is used to tune instruments in an orchestra to a pitch of exactly 440 Hz, which is the note A above middle C, in musical terms. When a tuning fork is struck against an object or surface, its two prongs, or *tines*, vibrate at their resonant frequencies and create sound waves that should correspond exactly to 440 Hz. An incorrectly designed tuning fork would cause instruments tuned with the device to be out of tune with other instruments, creating unwanted clashes in the harmonies you hear during a concert.

*Vibrating tuning fork geometry showing the acoustic pressure field and displacement of the fork’s prongs.*

Let’s take a look at an example of an analysis of a tuning fork in COMSOL (note that you can download this tuning fork model file below). This model has two basic steps of analysis:

- First, in
*Study 1*of the model, we analyze how the fundamental eigenfrequency of this particular shape of tuning fork changes as we vary the length of the prongs. We plot both the shape of the deformation of the fork as it vibrates, and we also plot the frequency the fork will naturally vibrate at when it is struck, based on the length of the prongs. You will see in this plot how the eigenfrequency drops as the prong length increases. The relationship between these two parameters is quite sensitive; an increase of only 1 mm in prong length corresponds to a decrease of about 10 Hz for the resonant frequency. This eigenfrequency vs. prong length plot below (left) shows us that the ideal prong length for this tuning fork is approximately 7.906 cm. - Now that we have determined our desired prong length, we want to investigate the interaction between the acoustics and the structural components of the fork to analyze the sound that will be radiating from the tuning fork out to the orchestra. We use a
*Frequency Domain*study type to see the characteristics of the sound produced for a harmonic structural load on a prong of the tuning fork that varies over frequency. From this analysis we are able to verify that the resonant frequency of this tuning fork is indeed at 440 Hz. This is depicted in the figure below (right). Note that only the relative levels are of interest here (their absolute values are inaudible), as we have simply applied a unit load in this example. We can also calculate the magnitude of the sound we would expect to hear from the tuning fork based on different magnitudes of applied structural loads. This study corresponds to finding the frequency response of the fork; that is, the*Fourier components*of the response. In practice, a tuning fork starts to vibrate when it is struck with another object, and from this object it receives an impulse consisting of several Fourier components. In this study we are also able to create plots of the radiation pattern of sound coming from the fork. This analysis gives us an accurate picture of how these two physics are working together to produce the tone heard by musicians while tuning their instruments in an orchestra.

*The left figure shows the fundamental vibrating frequency of the tuning fork as a function of the prong length L. The right figure depicts the sound pressure level measured in the proximity of the tuning fork (blue line) and measured at 1 m away from the fork (green line) at 440 Hz. Only the relative value is of interest as the fork is actuated by a unit load (right). Click to view larger images.*

Note that a theoretical solution exists that relates the resonate frequency of an “ideal” tuning fork to the other parameters of the fork, such as the fork material’s *Young’s modulus*, the material’s density, the radius of the prong cross sections, and the length of the two prongs. However, this theoretical solution assumes that the prongs of the fork are perfect cantilever beams, where in reality the bending stiffness of the prongs increases near the base of the fork where the prongs meet. Therefore, you should use COMSOL Multiphysics to explicitly model applications like the tuning fork where the real device you want to model doesn’t match up perfectly with the assumptions made by known theoretical solutions.

We find in this model that using theory alone we would arrive at a tuning fork with prongs of length 7.8 cm, but our COMSOL model has shown us that the real length needed for this particular tuning fork design is 7.906 cm. If we would have designed this tuning fork with the theoretically-determined prong length, our tuning fork would create a tone at about 430 Hz, which is nearly a quartertone away from the desired note A at 440 Hz. This difference would be very noticeable to the average listener and would create quite an undesirable clash of harmonies within the orchestra.

- Download the Tuning Fork with Acoustics model from the Model Gallery
- Or download the pure structural mechanics model of a the tuning fork geometry

Pipe organs create musical notes as a result of air being pushed through the pipes in a deliberate fashion. In addition to the pipes, every organ needs a component to force the air through, and a way of directing the air so the correct pipes are played. Each pipe can only play one note, which is determined by all of the structure’s elements. When it comes to getting a clear sound with great resonance, the pipe’s structural design needs to be finely tuned. In acoustics, pitch occupies a very precise mathematical space. If the sound shifts from the intended note, it will come through in the music and may degrade the sound quality. You can model and optimize the design of an organ pipe using COMSOL Multiphysics along with the Pipe Flow Module and Acoustics Module. It is helpful to be able to model an organ pipe *before* its fabrication and to plan for all of the specific parameters that come together to create great sound.

*Sketch of an organ pipe including the mouth and the pipe body.*

In the organ pipe design model in our Model Gallery, all of the parameters of a working organ pipe are depicted. The organ pipe walls require elastic qualities in order to be modeled correctly, enabling them to vibrate when sound is propagating through them. Different wall properties slightly alter the perceived speed of sound in the pipes. The pipe in this particular model works at 440 Hz to produce the note A4 (or a’). The air flows in through the bottom of the pipe, up along the body, and out through the top where the mouth is. The mouth of the organ pipe has a carefully designed upper lip system, where the turbulence at the outlet initiates vibrations. The vibrating pipe body causes the pipe’s note to sound. The fundamental tone of the note corresponds to a half wave resonance in the pipe. Resonances of shorter wavelength generate harmonics of the fundamental frequency. All of the harmonics combined with the fundamental tone determine the exact pitch of the organ pipe.

*Resonance peak of the fundamental frequency at 440 Hz for different inner pipe radii.*

Any slight change in the parameters (such as pipe radius, shown above) will result in changes in the damping (the acoustic impedance at the open end of the pipe will change) and thus change the Q value of the frequency response resonance peaks of the organ pipe. This will then create a different pitch. The acoustic properties of the system can easily be studied with respect to other parameters, such as the pipe cross section shape, the pipe material, or the steady flow of air in the pipe. For example, a slight residual airflow *u*_{0} in the pipe could slightly change the resonance frequency.

The pipe geometry for this particular model is defined with length L, inner pipe radius *a*, wall thickness *dw*, and a circular cross section shape. Changing the width of the pipe wall will slightly change the resonance, as shown in the following graph. The resonance changes because the elastic properties of the pipe wall determine how compressible a certain cross section is. The width of the pipe wall impacts the speed of sound in the pipe, which causes the resonance to change as well.

*Resonance peak of the fundamental frequency at 440 Hz for different pipe wall thickness.*

As designers tweak and alter the materials and exact parameters of the organ pipe design, they are able to improve the sound of the instrument. The technology behind acoustics and pipe design has improved vastly over the years, and the sound quality continues to become more crisp and powerful. The tools that instrument designers have at their disposal are also always improving.

*Propagation of sound waves from a sound source.*

As fans of the popular TV-show, *The Big Bang Theory*, have heard it described, the Doppler effect is the apparent change in the frequency of a wave caused by relative motion between the source of the wave and the observer. But what exactly does this mean?

One of the easiest ways to visualize the Doppler effect is to imagine a bug gliding across the surface of a puddle. First, let’s picture what the disturbances in the puddle would look like if the bug was stationary, vibrating its legs and producing waves in the puddle. The disturbances would propagate outward from the bug in spherical waves, resembling what is seen in the image above. However, what would happen to these waves if the bug started moving across the water? The water flow around the bug would change so that the waves are closer together in front of the bug, and farther apart behind it. This can be seen in the following animation:

*Doppler effect animation showing waves emitted from
an object moving to the right. Animation attribution:
Lookang.*

The Doppler effect works in much the same way for sound. When a sound source is stationary, the sound that we hear is the same that is emitted from the sound source. However, when the sound source begins to move, the perceived sound changes. Let’s use the ambulance example again. Not only is the sound that we hear different for an ambulance moving away from us, but the sound reaching our ears is different as the ambulance approaches, when it is parallel to our location, and as it recedes.

In the first case, as the ambulance moves toward us, each successive sound wave is emitted from a closer position than that of the previous wave. Because of this change in position, each sound wave takes less time to reach us than the previous one. The distance between wave crests (the wavelength) is thereby reduced, meaning that the perceived frequency of the wave is increased. The sound is perceived to be of a higher pitch. Conversely, as a sound source moves away, waves are emitted from a source that is farther and farther away, therefore creating an increased wavelength, a decreased perceived frequency, and a lower pitch.

We can use COMSOL Multiphysics and the Acoustics Module to create a simulation of the Doppler effect to measure the change in frequency for a sound source moving at a certain velocity. In our simulation, let’s assume that the air surrounding the sound source (the ambulance in this case) is moving with a velocity of V = 50 m/s in the negative *z*-direction. We’ll also assume that the observer of the sound is standing 1 m from the ambulance as it passes by. In the figure below, we can see the change in sound pressure level as the ambulance approaches and passes an observer:

*A Doppler effect simulation where the distance of the ambulance from the observer is represented on the x-axis.
The solid line represents the pressure perceived by the observer of an approaching ambulance. The dashed line
shows the pressure as the ambulance recedes.*

From this plot, we can see how the amplitude of the wave (or pressure) drops off at a faster rate when the ambulance is moving away from an observer compared to when it is approaching. The change in the amplitude of the wave depicts how the siren’s sound becomes quieter as the ambulance moves away. The rate at which the sound level decreases as the ambulance recedes is much faster than the rate at which the sound becomes louder as the ambulance approaches, as can be seen in the graph. We can also visualize the sound pressure level around the sound source:

*Sound pressure level around the sound source represented by colors and contour lines. You can see how the
outermost contour runs from well inside the physical domain to the PML, showing that the sound is greater
below than above the source.*

The Doppler effect can also be seen in many other phenomena. One common example is the Doppler radar, where a radar beam is fired at a moving target. The time it takes for the radar to bounce off the target and return to the transmitter can provide information about a target’s velocity. Another example is in astronomy, where the Doppler effect is used to determine the direction and rate at which a star, planet, or galaxy is moving compared to the Earth. By measuring the change in the color of electromagnetic waves — called redshift or blueshift — an astronomer can determine the celestial bodies’ radial velocity. Other applications that take advantage of the Doppler effect include sonar, medical imaging and blood flow measurement, satellite communication, and many more.

- Download the Doppler Shift model from the Model Gallery