# How to determine the type of wave and wave speed in a membrane type material?

I have a PET foil and I attach it to a frame with sensors pasted on the foil edges to detect impacts on the foil. Since it is a really thin material, I categorize it as a membrane and not as a bulk solid object. But a bunch of things regarding the basic physics are still unclear to me and they are as follows:

1. How do I know what type of wave is travelling when the impact happens ? would it be longitudinal or tranverse ?
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.
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.
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.
5. Can I say that if I have an acoustic wave then it is a longitudinal transmission of waves ?
6. Is it possible that I have a combination of longitudinal and transverse waves when the impacts happen ?
7. Do the waves have different speeds of propogation in x and y direction ? Is there an equation to determine this ?
8. Can I only determine the speed of waves in a material if I have a standing wave ? experimentally or theoretically
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 ?
10. Is there a difference in formula when calculating wave speed for a solid vs a membrane type of material ?

It would be really helpful if these questions get clarified because I am totally confused.

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.

• Thank you for the answers, I've been searching about them for months. However I have some follow up questions. In pt 2 you say thin foils have none of the shear or compressive waves, so what kind of waves do they have ? Don't you mention in pt 1 that it is actually a mixture of these both ? Nov 28, 2021 at 10:53
• In pt 6, how are you so sure that it is definitely a combination of longitudinal and transverse waves happening ? Nov 28, 2021 at 10:56
• From pt 7, is there an equation to determine the dependence of membrane tension on wave speed ? Nov 28, 2021 at 10:59
• My comment in pt 2 was that you can't isolate the two wave speeds in a membrane, not that they don't exist. The vibration wave in a membrane is a combination of the the two and you can't extract one from the other. My comment in pt 6 is based on the theory of membranes; the theory simplifies the real situation in such a way that you lose the ability to model anything but the combination. In appropriate situations this approximation is very good. Finally, there are equations but a full discussion is outside the scope of this question; try reading "Fundamentals of Acoustics" by Kinsler and Frey. Nov 29, 2021 at 14:44