# How does acoustic impedance change with frequency of the wave and thickness of the medium?

I have read in many posts that the acoustic impedance of a material is usually defined as $$Z = pc$$, where $$p$$ is the density of the material under consideration and $$c$$ is the speed of sound in that material. However, this is the "Characteristic specific acoustic impedance" as explained here. I wonder how the acoustic impedance of a material is related to its dimensions, especially the thickness.

For instance, if I have one atom thick layer of copper, does this offer the same acoustic impedance of a one meter thick layer of copper? I guess that in that case I will have to refer to the more general law. Moreover, how does the acoustic impendace change with the frequency of the acoustic wave? E.g. Does a $$100\;Hz$$ acoustic wave that is hitting a certain material see the same acoustic impedance of a $$1\;MHz$$ acoustic wave hitting the same material?

• Hi Gabriele, welcome to the Physics SE. Please use the MathJax syntax for better readability of the formulas in your post. Mar 14, 2019 at 17:18

The acoustic impedance of an atom is meaningless. The acoustic impedance describes the response of a medium to the propagation of some acoustic excitation. Are you implying a sound wave that propagates through an atom? Anyway, the classical acoustic theory is developed for continuous media so you cannot expect it to work at atomic scale.

Now, within the limits of macroscopic media, the acoustic impedance can be defined for a medium (what you mentioned as specific impedance) or for an object (for example, the membrane of a speaker). The specific impedance does not depend on the geometry of the medium (like density does not depend on the size of the object). But the impedance of the objects of course it's a function of size and shape. It may depend on frequency too (for speaker membranes for example it does, in general).

If you want to know the specific variation of impedance with geometry and frequency you need to look up some research papers for the specific geometry. You may find relatively easily the results for a disk (or piston) which is one the usual examples in acoustics textbooks.

• Sound travels though monolayer graphene just fine, and it more or less like an ideal membrane on acoustic length scales. I wouldn't call the impedance of a monolayer meaningless. May 28, 2019 at 2:37
• Did I say that the impedance of a membrane is meaningless?
– nasu
May 28, 2019 at 11:23
• @KFGauss, you could probably define a "continuum" in a monolayer of graphene. Also, you probably need a generalized theory such as micropolar elasticity instead of classical elasticity. Sep 8, 2021 at 21:58
• I agree that membranes can sustain mechanical waves. I suppose I read the OP as asking for a single atom whereas he talks about an atom thick membrane. The impedance of the membrane will depend on its geometry, boundary conditions and elastic properties.
– nasu
Sep 8, 2021 at 23:13