Limits for the linear wave equation

Question

In acoustics and continuum mechanics the following wave equation (for Speed of Sound $c$) for the pressure field $p$ is well-known: $\partial_t \partial_t p = c^2 \Delta p$.

This wave equation can be obtained by combining the equation of continuity and the Euler equation for fluids and linearization of the pressure, density and velocity fields. The linearization makes sense because the Sound pressure is much smaller than the atmospheric pressure. Question: In which cases the acoustic wave equation must be Extended?

I think the acoustic wave equation has to be Extended if the Sound pressure is high (very loud Events?). Considering the sound of a Train horn (approx. 140dB(A)) in a small neighborhood (3 feet). Is it necessary to describe the Sound Propagation with nonlinear wave equations?

Answer 1

Let's review the linearisation and go to the further details. Just the pressure might be not enough.

Take the momentum equation:

$$ -\frac{1}{\rho}\nabla p = \frac{\partial \vec{v}}{\partial t}+\vec{v}\cdot\nabla\vec{v} $$

Here we have to eliminate the convective part $\vec{v}\cdot\nabla\vec{v}$. Usually the argumentation is that changes of the velocity are small (i.e. the product is small of the 2nd order) and we are in the inertial system connected to the fluid (i.e. the mean flow is zero).
More to that, if the viscosity should be added (e.g. due to the friction on the boundaries) the $\nabla p$ is not enough to describe the tensions in the fluid.

Continuity equation:

$$ \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\vec{v})=0$$

We assume $\rho(r,t)=\rho_0+\rho'(r,t)$ and $\rho_0>>\rho'$ to get the linearised equation. Therefore, "too big variation of density is not allowed".

Energy equation - we assume the adiabatic (and therefore homentropic!!!) behavior. When the changes in enthropy occurs (e.g. again the boundary layers), this simple system is not valid anymore.

So, don't limit your considerations just to pressure.

To your example with train horn: There is one more thing to add - the difference between the near field and far field. The system tends to be more linear at greater distances from the horn and it's the science itself to make a proper discussion on the linear/non-linear behavior. But I think 3 feet are moreless enough to consider the system linear.