Acoustics

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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:

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

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 $\underline v$ 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 $?$ is:

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

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.