I used to work on fast algorithms for solving acoustical wave equation, e.g., as explained in https://en.wikipedia.org/wiki/Wave_equation. Now I am working in a semiconductor company, and we study optics. As I searched, e.g., as explained in http://faculty.washington.edu/lylin/EE485W04/Ch2.pdf, I didn't find much difference between acoustical wave equation and optical wave equation, except the difference between speed of sound and speed of light. As we know, in acoustics, wave equation is derived from conservation of mass and conservation of momentum of the medium of sound propagation, gas. Does light have a similar medium that satisfies conservation of mass and conservation of momentum? Does it mean an acoustic solver can be applied to studying optics just by changing the speed of sound to speed of light?
Physics – Difference Between Optical Wave Equation and Acoustics Wave Equation
acousticsoptics
Related Solutions
To solve a non homogeneous wave equation you need to use Green's function. The Green function of the inhomogeneous wave equation is defined as $$ (\nabla^2-\frac{1}{c^2}\partial_t^2)G(x,x';t,t') =\delta(x-x')\delta(t-t') $$ It means that $G$ is the solution of the wave equation for a Dirac source localized at the position $r'$ and at a time $t'$. Why is this Green function important ? Because once you have it you can have any solution $\Phi(x,t)$ to the equation $$ (\nabla^2-\frac{1}{c^2}\partial_t^2)\Phi(x,t)=\rho(x,t) $$ by computing $$ \Phi=\int G(x,x';t,t')\rho(x',t')dx'dt' \space \space\space\space\space(1) $$
because
$$
(\nabla^2-\frac{1}{c^2}\partial_t^2)\Phi=\int (\nabla^2-\frac{1}{c^2}\partial_t^2)G(x,x';t,t')\rho(x',t')dx'dt'
$$
using the definition of the Green function:
$$
=\int \delta(x-x')\delta(t-t')\rho(x',t')dx'dt'= \rho(x,t)
$$
In equation (1) lies the answer to your question: To obtain your solution you have to sum up the influence of every infinitesimal part of your source term $\rho$. This influence is given by the Green function, which is exactly the solution of the same problem for an infinitesimal pointlike source.
Its a little bit simpler in frequency domain, or even in static (where you have Poisson or Laplace equation instead of the wave equation) but the idea is always the same: obtain a Green function for a point-like (Dirac source) and then obtain the solution by integration.
A good example to physically illustrate this idea is the Huygens principle: every part of the phase front of a wave is a little source of a spherical wave. If you want the solution of a problem where your source ($\rho$) is a circular hole cut in an opaque screen you'll have to sum up little spherical wave (Green function) over all the hole surface. That's how diffraction problem are solved.
1. How do I know what type of wave is travelling when the impact happens? Would it be longitudinal or transverse?
For very thin solids usually you will have a mixture of longitudinal and transverse motion occurring. It would be inaccurate to try and classify the motion of a membrane as either case individually.
2. Is the speed of sound in the material same thing as wave speed in the material? Sometimes it is used interchangeably and it is confusing.
Calling mechanical wave motion in solids "sound" is a little confusing, as we typically associate sound with something we hear, which intrinsically is associated with fluids. However, for the most part you can use "wave speed" and "sound speed" interchangeably even for solids (unless they are specifying a non-mechanical wave). It is important to remember that there are two different wave speeds in bulk solids (compressive and shear), and neither of these are the wave speed for thin membranes.
3. Are waves created due to impacts always acoustic waves? I think the waves in the foil might be acoustic since it travels through a medium and I can hear it but not see it.
Similarly with the discussion for "sound", "acoustic" can be used to describe small-amplitude (no nonlinear effects) mechanical vibrations in solids as well as fluids.
4. Are acoustic waves a type of longitudinal wave or is the the other way around ? I could not find a proper classification tree for the wave types.
If someone is trying to make a distinction, acoustic refers to longitudinal (compressional). However, there are plenty of cases when "acoustic" can also refer to transverse (shear) waves.
5. Can I say that if I have an acoustic wave then it is a longitudinal transmission of waves?
See my response to 4.
6. Is it possible that I have a combination of longitudinal and transverse waves when the impacts happen?
Not only is it possible, it is almost certainly what is happening.
7. Do the waves have different speeds of propagation in x and y direction? Is there an equation to determine this?
In the bulk, this only happens if the material is different in one direction than another (think of wood along the grain versus against the grain). In a membrane you get a lot of your wave motion from the tension of the membrane; if you pulled one side tighter than the other, the wave speeds will be different in the two directions.
8. Can I only determine the speed of waves in a material if I have a standing wave? experimentally or theoretically
No, you can readily determine the speed of waves using propagating, non-standing, waves. One approach that might be helpful is to measure a propagating signal at two points and do a cross-correlation between the two signals. The peak in the time difference gives you how long it took the wave to get from the one measurement site to another. However, since you have a bounded domain, my guess is that you will wind up using standing waves if you are doing this experimentally.
Theoretically, if the foil is truly thin enough to be considered a membrane (which it probably is), then you can just use the formula for the wave speed in a taut membrane: $$c = \sqrt{T/\sigma},$$ where $c$ is the wave speed, $T$ is the tension of your membrane and $\sigma$ is the areal mass density (mass density of your material times the membrane thickness).
9. To calculate the wave speed using the formula of $\sqrt{B/\rho}$, is it necessary that I know the type of wave that is travelling in my medium? That is whether it is longitudinal or transverse?
That expression is a generic expression for waves in a bulk, where $B$ could be the shear modulus for transverse waves or it could be a mixture of shear and bulk moduli for compressional waves. You need to know what you are working with to use it appropriately. But, remember that waves on a membrane are not in the bulk, so don't use this equation for your case at all. See the answer to 8.
10. Is there a difference in formula when calculating wave speed for a solid vs a membrane type of material ?
Yes; see my answer to 8.
Best Answer
The "wave equation" is the same for both but the acoustic is a scalar while the EM is vectorial type. Depending on the medium and/or scattering obstacles the several components (6 = 3 Electric field + 3 Magnetic field) may interact and then it becomes a lot more complicated problem than just the scalar case.
Per @Jagerber48, if the medium is simple (homogeneous, isotropic and lossless) and interactions between the components at the scattering boundaries are negligible then the vector equation separates into scalar equations and the acoustic solver should work for you. EM component mixing, ie., the vector behavior, is important if you want to see sub-wavelength details of the field very near the scattering object.