# Acoustics/Fundamentals of Acoustics

![](Acoustics_Fundamentals_of_Acoustics.jpg "Acoustics_Fundamentals_of_Acoustics.jpg")

## 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 equation (see
wave equation):

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```
$\nabla ^2 p - \frac{1}{{c_0 ^2 }}\frac{{\partial ^2 p}}{{\partial t^2 }} = 0$

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</div>
```
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, $c_0$ 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:

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

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

## Helmholtz equation

Since the velocity field $\underline v$ for acoustic waves is
irrotational we can define an acoustic potential $\Phi$ by:

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```
$\underline v  = \text{grad }\Phi$

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

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

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</div>
```
Applying the Fourier Transform, we get the widely used Helmholtz
equation:

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```
$\nabla ^2 \hat \Phi  + k^2 \hat \Phi  = 0$

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</div>
```
where $k$ is the wave number associated with $\Phi$. 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:

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```
$\underline i (t) = p\underline v$

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```
We can then define the average intensity:

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```
$\underline I  =  \langle \underline i  \rangle$

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```
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:

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

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

## 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:

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

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</div>
```
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:

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```
$\frac{p}{v} = Z =  \pm \rho _0 c_0$

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

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```
$\underline i  =  \pm \frac{{p^2 }}{{\rho _0 c_0 }}\underline {e_x }$

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</div>
```
### Spherical waves

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

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

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</div>
```
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.

