# Does Fermat's principle of least time apply to sound waves?

I am reading Feynman's presentation of Fermat's Principle of Least Time, which explains the behavior of light; does it apply to waves in general? for example sound waves or waves on the surface of water?

In particular does Feynman's method of summing probability amplitudes apply to sound waves? (naturally we would have to talk about amplitude of disturbance rather than amplitude of probability) ## 2 Answers

Yes. Fermat's principle is equivalent to the Eikonal Equation, which can be derived for any system fulfilling the D'Alembert wave equation when a slowly varying envelope approximation is valid. For approximately monotonal sound waves of wavenumber $k$, the Helmholtz equation with position dependent sound velocity $(\nabla^2+k^2(\vec{r}))\psi= 0$ is approximately fulfilled. We make an Ansatz of the form $\psi(\vec{r}) = A(\vec{r}) \exp(i\,\phi(\vec{r}))$. On putting this Ansatz into Helmholtz's equation we get:

$$(\nabla^2 A + 2 i \nabla A\cdot \nabla\phi + i A \nabla^2 \phi-A |\nabla\phi|^2 + k^2(\vec{r}) A )\exp(i\,\phi(\vec{r})) = 0$$

In a slowly varying envelope approximation, the first two terms are very small compared to the rest. We also approximate $\nabla^2 \phi$ as $k_0^2$, where $k_0$ is some representative wavenumber, so that we are left with the Eikonal equation $|\nabla\phi|^2 = k^2(\vec{r})-k_0^2$. This then implies Fermat's principle, as discussed on the Eikonal Equation Wikipedia Page.

Hamilton's principal of least action would apply which is more general than Fermat's principle. If Energy remained constant then action would be proportional to time so probably something like Fermat' principle would apply, but this is just a guess.

• It is not clear to me how Hamilton's principle can be applied to the problem, since Fermat's principle doesn't require a single path but a summation over all possible paths, for example reflection by a non standard angle from a mirror with particular pattern of blacked out lines, etc... – nir Apr 24 '15 at 21:03