1.1: Fundamentals of Acoustics

Introduction

Sound is an oscillation of pressure transmitted through a gas, liquid, or solid in the form of a traveling wave, and can be generated by any localized pressure variation in a medium. An easy way to understand how sound propagates is to consider that space can be divided into thin layers. The vibration (the successive compression and relaxation) of these layers, at a certain velocity, enables the sound to propagate, hence producing a wave. The speed of sound depends on the compressibility and density of the medium.

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

Equation of waves

Sound waves consist in the propagation of a scalar quantity, acoustic over-pressure. The propagation of sound waves in a stationary medium (e.g. still air or water) is governed by the following wave equation:

\[{\displaystyle \nabla ^{2}p-{\frac {1}{c_{0}^{2}}}{\frac {\partial ^{2}p}{\partial t^{2}}}=0}\]

This equation is obtained using the conservation equations (mass, momentum and energy) and the thermodynamic equations of state of an ideal gas (or of an ideally compressible solid or liquid), supposing that the pressure variations are small, and neglecting viscosity and thermal conduction, which would give other terms, accounting for sound attenuation.

In the propagation equation of sound waves, \({\displaystyle c_{0}}\) 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:

\[{\displaystyle c_{0}={\frac {1}{\sqrt {\rho _{0}\chi _{s}}}}}\]

where \({\displaystyle \rho _{0}}\) is the density and χS\({\displaystyle \chi _{S}}\) is the compressibility coefficient of the propagation medium.

Helmholtz equation

Since the velocity field v_\({\displaystyle {\underline {v}}}\) for acoustic waves is irrotational we can define an acoustic potential Φ\({\displaystyle \Phi }\) by:

\[ {\underline {v}}={\text{grad }}\Phi \]

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

\[ \nabla ^{2}\Phi -{\frac {1}{c_{0}^{2}}}{\frac {\partial ^{2}\Phi }{\partial t^{2}}}=0\]

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

\[\nabla ^{2}{\hat {\Phi }}+k^{2}{\hat {\Phi }}=0\]

where \({\displaystyle k}\) is the wave number associated with \({\displaystyle \Phi }\). 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:

\[{\displaystyle {\underline {i}}(t)=p{\underline {v}}}\]

We can then define the average intensity:

\[{\displaystyle {\underline {I}}=\langle {\underline {i}}\rangle }\]

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:

\[p^{\rm {dB}}=20\log \left({\frac {p}{p_{\mathrm {ref} }}}\right)\]

or acoustic average intensity:

\[L_{I}=10\log \left({\frac {I}{I_{\mathrm {ref} }}}\right)\]

where \({\displaystyle p_{\mathrm {ref} }=2*10^{-5}~{\rm {Pa}}}\) for air, or \({\displaystyle p_{\mathrm {ref} }=10^{-6}~{\rm {Pa}}}\) for any other media, and \({\displaystyle I_{\mathrm {ref} }=10^{-12}~{\rm {W/m}}^{2}}\).

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:

\[\Phi (x,t)= f\left(t- \frac {x}{c_{0}} \right) + g\left(t+ \frac {x}{c_{0}} \right)\]

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 \(\underline {v}\) 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:

\[ {\underline {i}}=\pm {\frac {p^{2}}{\rho _{0}c_{0}}}{\underline {e_{x}}}\]

Spherical waves

More generally, the waves propagate in any direction and are spherical waves. In these cases, the solution for the acoustic potential \({\displaystyle \Phi }\) is:

\[ \Phi (r,t)={\frac {1}{r}}f\left(t-{\frac {r}{c_{0}}}\right)+{\frac {1}{r}}g\left(t+{\frac {r}{c_{0}}}\right)\]

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, it is simpler 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 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.