Questions tagged [shock-waves]

A shock wave results from a nonlinearly steepening compressional mode that has reached a balance between nonlinear steepening and energy dissipation. Nonlinear wave steepening arises from the the $\left( \mathbf{V} \cdot \nabla \right) \mathbf{V}$ term -- often called the nonlinear or steepening term -- in the equations of motion. Energy dissipation is the transformation of energy from one form (e.g., bulk kinetic energy) to another (e.g., heat) through an irreversible process like wave dispersion (e.g., $\propto \ \beta \ \partial_{x}^{3} v$), viscosity (e.g., $\propto \ \nu \ \partial_{x}^{2} v$), friction (e.g., $\propto \ \zeta \ v$), etc. Generally it is assumed that the disturbance which drives the compression (often referred to as a piston, though it need not be an actual piston) move faster than the relevant speed of communication in the given medium, e.g., the speed of sound in Earth's atmosphere, for a shock wave to form.

Nonlinear wave steepening is the process by which a wave front can change its profile because the wave phase speed depends upon the wave amplitude. For instance, conceptually one can say that the peak of a wave will propagate faster than the trough, resulting in a pile up in the direction of propagation. A sound wave is a compressive mode that behaves in such a way. In fact, were it not for dissipative effects, a sound wave produced by a spoken word would steepen until it either broke or formed a shock, depending upon whether dissipation balanced steepening.

In summary, there are a few points to consider:

1. If the dissipative effects dominate over the nonlinear term, then the wave will not steepen.
2. If the nonlinear term dominates the dissipative term(s), then the compressive fluctuation will steepen until it reaches a gradient catastrophe where it becomes a multi-valued function. This is also known as wave breaking, which is often seen in surface water waves.
3. If the dissipative effects balance the nonlinear term and the disturbance propagates at least as fast as the relevant communication speed, then a shock wave can form.