Acoustics/Fundamentals of Acoustics

Introduction

Sound is due to any variation in the atmospheric pressure. An easy way to understand how sound propagates is to consider that space can be divided into thin air layers. The vibration (the successive compression and relaxation) of these layers, at a certain velocity, enables the sound to propagate, hence producing a wave. This is the reason why sound waves cannot exist in an incompressible fluid.

In this chapter, we will only consider the propagation of sound waves in an area without any acoustic source, in an homogeneous fluid.

Equation of waves

Sound waves consist in the propagation of a scalar field, acoustic over-pressure, and a vector field, acoustic local velocity. Thus, the propagation of sound waves is governed by the two following equations, which are equivalent:

${\mathrm{\nabla }}^{2}p-\frac{1}{c_0^2}\frac{{\partial }^{2}p}{\partial x^2}=0$  ; ${\mathrm{\nabla }}^2\underset{\_}{v}-\frac{1}{c_0^2}\frac{{\partial }^2\underset{\_}{v}}{\partial x^2}=0$

These equations are obtained using the conservation equations (mass, momentum and energy), the thermodynamic equations of state as well as behavior laws (Newtonian fluid, Fourier’s law of conduction). Once viscosity and conduction have been neglected, we consider that all perturbations remain small enough for the previous equations to be linearized (for example, the non-linear term in the momentum equation can be neglected). For specific cases, where acoustic over-pressure becomes too high (sonic boom, etc.), all non-linear terms must be kept, and we then have to deal with non-linear acoustics.

In the propagation equation of sound waves, $c_0$ is the propagation velocity of the sound wave (which has nothing to do with the vibration velocity of the air layers). This propagation velocity has the following expression:

$c_0=\frac{1}{\sqrt{{\rho }_0{\chi }_s}}$

where ${\rho }_0$ is the density and ${\chi }_S$ is the compressibility coefficient of the propagation medium.

Helmholtz equation

Since the velocity field $\underset{\_}{v}$ for acoustic waves is irrotational we can define an acoustic potential $\mathrm{\Phi }$ by:

$\underset{\_}{v}=grad\mathrm{\Phi }$

Using the propagation equation of the previous paragraph, it is easy to obtain the new equation:

${\mathrm{\nabla }}^2\mathrm{\Phi }-\frac{1}{c_0^2}\frac{{\partial }^2\mathrm{\Phi }}{\partial t^2}=0$

Applying the Fourier Transform, we get the widely used Helmoltz equation:

${\mathrm{\nabla }}^2\hat{\mathrm{\Phi }}+k^2\hat{\mathrm{\Phi }}=0$

where $k$ is the wave number associated with $\mathrm{\Phi }$. Using this equation is often the easiest way to solve acoustical problems.

Acoustic intensity and decibel

The acoustic intensity represents the acoustic energy flux associated with the wave propagation:

$\underset{\_}{i}(t)=p\underset{\_}{v}$

We can then define the average intensity:

$\underset{\_}{I}=<\underset{\_}{i}>$

However, acoustic intensity does not give a good idea of the sound level, since the sensitivity of our ears is logarithmic. Therefore we define decibels, either using acoustic over-pressure or acoustic average intensity:

$p^{dB}=20\log (\frac{p}{p_{ref}})$  ; $L_I=10\log (\frac{I}{I_{ref}})$

where $p_{ref}=2.10^{-5}Pa$ for air, or $p_{ref}=10^{-6}Pa$ for any other media, and $I_{ref}=10^{-12}$ W/m².

Solving the wave equation

Plane waves

If we study the propagation of a sound wave, far from the acoustic source, it can be considered as a plane 1D wave. If the direction of propagation is along the x axis, the solution is:

$\mathrm{\Phi }(x,t)=f(t-\frac{x}{c_0})+g(t+\frac{x}{c_0})$

where f and g can be any function. f describes the wave motion toward increasing x, whereas g describes the motion toward decreasing x.

The momentum equation provides a relation between $p$ and $\underset{\_}{v}$ which leads to the expression of the specific impedance, defined as follows:

$\frac{p}{v}=Z=\pm {\rho }_0{c}_0$

And still in the case of a plane wave, we get the following expression for the acoustic intensity:

$\underset{\_}{i}=\pm \frac{p^2}{{\rho }_0{c}_0}\underset{\_}{e_x}$

Spherical waves

More generally, the waves propagates in any direction and are spherical waves. In these cases, the solution for the acoustic potential $\mathrm{\Phi }$ is:

$\mathrm{\Phi }(r,t)=\frac{1}{r}f(t-\frac{r}{c_0})+\frac{1}{r}g(t+\frac{r}{c_0})$

The fact that the potential decreases linearly while the distance to the source rises is just a consequence of the conservation of energy. For spherical waves, we can also easily calculate the specific impedance as well as the acoustic intensity.

Boundary conditions

Concerning the boundary conditions which are used for solving the wave equation, we can distinguish two situations. If the medium is not absorptive, the boundary conditions are established using the usual equations for mechanics. But in the situation of an absorptive material, we cannot write the equations for mechanics, and therefore, we have to use the concept of acoustic impedance.

Non-absorptive material

In that case, we get explicit boundary conditions either on stresses and on velocities at the interface. These conditions depend on whether the media are solids, inviscid or viscous fluids.

Absorptive material

Here, we do not know the equations for mechanics in the absorptive material and thus, we have to use the acoustic impedance as the boundary condition. This impedance, which is often given by experimental measurements depends on the material, the fluid and the frequency of the sound wave.