This blog post has previously been posted on LinkedIn, but since such content requires an account, it has been reposted here.

#### Introduction

When investigating the sound pressure from a loudspeaker driver, the displacement of the diaphragm has to be either known, e.g. via an electro-acoustic analogy circuit, and then an analytical expression can be used for the subsequent pressure calculation, or the displacement can be calculated/coupled to the surrounding air using e.g. the finite element method. Mechanical losses in the loudspeaker driver have to be included, usually via some fitting to measurements, to have accurate results, but it is not customary, nor necessary (except for special drivers), to include any acoustic losses.

For room acoustics, losses are at the most included as one or more boundary conditions, that may have to be fitted to the room in question, but there are rarely any explicite calculation of the losses on the boundaries.

For many applications/technology areas, acoustic losses are not very important, and it suffices to stick with 'standard acoustics', i.e. where the processes involved in the sound generation are adiabatic and reversible, and so the energy is contained within the pressure field. However, for small geometries such as tubes and slits with narrow cross-sections, the so-called thermoviscous effects are of great importance, as they are responsible for the losses in the acoustic system. One application where accurate calculation results depend on including these losses is a hearing aid, but for microphones and other small tranducers the effects can be equially important.

The theory behind thermoviscous losses is described very concisely here by Mads Herring Jensen, Tecnical Product Manager, Acoustics, COMSOL, and it is recommended both for giving an overview and also for going more into the equations, if needed. For now, what is important to know that the thermoviscous effects take place in a boundary layer with a frequency-dependent thickness; as the frequency increases, the thickness becomes smaller. In the audio range the thickness varies from 0.5 mm to 0.015 mm. The ratio between these two numbers is only about 33 over two decades, since the boundary layer thickness is proportional to the square root of frequency.

When an application has dimensions in the millimeter/sub-millimeter range, it should be investigated if the loss effects are of importance. Not knowing the exact working range of the modeling methods can easily somewhat obscure an otherwise great publication, and a recent motivation for including acoustic losses in calculations is the rising amount of research being done in acoustic topology optimization: Here, lossless and often 2D topology optimization is carried out, and if a slab of the optimized geometry are then later 3D printed with a short depth dimension, a substantial amount of loss is added, and the result no longer looks so optimal. Also, much attention is being given to metamaterial constructions, where certain acoustic properties, such as acoustic cloacking, is sought mathematically without including losses, and again measurement results often look less convicing than original expected.

Below is my take on the most common modeling methods for thermoviscous losses in acoustics.

#### Lumped parameter model

The simplest method assumes that each part of the acoustics system can be lumped into a single component. This is known for electro-acoustics analogy circuits, where acoustic masses and compliances are input as individual components. The general lumped method assumes that:

The acoustic wavelength is much larger than any characteristic length in any of the lumped geometries.

This is sometimes described as "the pressure must be constant within each lumped geometry" in the literature, but this is not entirely correct; for cavities the constant variable is the pressure, but for masses it is the velocity (proportional to the pressure gradient) which is constant.

When thermoviscous losses are included it is additional assumed that:

The acoustic wavelength is so large that the boundary layers are small compared to it.

This is related to the fact that at low frequencies the viscous and thermal losses are constant, whereas for higher frequencies they become frequency-dependent. At these higher frequencies the lumped component values are not constant. However, with modern software this may not be a problem at all.

The general method is usually denoted as a zero-dimensional method, which is true in the sense that each component has no variation of its associated acoustic variable in a specific length direction. On the other hand, the acoustic impedances are found on the basis of knowing a characteristic area and a length, which implies a certain 1D characteristic: This becomes especially apparent when the lossy elements are included, since the value of the loss is highly dependent on the shape of the circumference of the geometry, hence a certain length direction is implied.

For me, it makes good sense to think of the lumped method as a special case of the next method, namely the:

#### Transmission line model

When acoustic wave propagation is allowed in one dimension only, then a transmission line modeling is usually ideal. The frequency has to kept low enough that the acoustic wavelength is larger than a characteristic cross-sectional length, which ensures plane wave propagation. For transmission lines plenty of theory exists for linking input pressure and velocity at one end of the transmission line with ditto variables at the output. In acoustics, the ABCD transfer matrix method is often employed, but other possibilities work equally well. The losses are included in the system via analytical expression, and this can generally only be done for certain cross-section geometries, such as e.g. circular cross-sections. For general cross-sections geometries other methods must be used.

When it comes to implementation in FEM software, where part of the system is modelled as a transmission line, constraining the system so that it becomes critically determined can be a little tricky, but some acoustic software packages have transmission line functionality directly built in.

#### Boundary layer impedance model

Since the viscous and thermal losses take place in a boundary layer, it would make sense to apply an impedance with a non-zero real part to include damping in the system. However, the boundary layer has a finite length which is largest at lower frequencies, and below a certain frequency, it is not valid to assume that the thermoviscous effects take place exactly on the boundary. Hence, the boundary layer impedance model only works for higher frequencies. Also, at very high frequencies, the losses tend to be overruled by momentum and compression effects anyway, so there is only a limited range for the boundary layer impedance to work within. There has, however, been some advances made in recent years with the method, but we will not touch further upon this here.

#### Low reduced frequency model

The low reduced frequency model is tightly linked with the transmission line model, as they usually work under similar assumptions. The low reduced frequency model is intended to have a finite element method implementation, with only pressure as a degree of freedom, hence knowing the analytical expressions linked to a certain cross-section will allow you to do fast calculation with accurate thermoviscous losses included via this 'numerical transmission line'. Much literature can be found on the low reduced frequency model, and it still gets attention and improvements in recent publications and on-going research.

#### Full linearized Navier-Stokes model

The full linearized Navier-Stokes model employs several coupled differential equations which combined describe the behavior of the pressure, velocity and temperature variation in the acoustic system. The method is not limited to any specific geometry, as opposed to the other methods described here, and so can be implemented in a finite element software. The only software to my knowledge which currently has this option is COMSOL Multiphysics. While very general, it can be very computationally heavy to solve the matrix systems related to the coupled equations. Also, meshing of the finite element domain must be done in such a fashion that the thermovisous effects are accurately captured.

#### Conclusion

As seen from the previous sections, several methods exist for evaluating thermoviscous losses, and additional methods have been developed in recent years*. These losses are only becoming of more and more interested with the growing focus on acoustic topology optimization and metamaterials, and so it is important to choose the method which at the one hand is accurate enough, and at the other hand is not too demanding when it comes to computation time and RAM requirements.

* I may update this article or make a part 2 with these methods.