In the same medium the sound intensity obeys the inverse square law, but what happen if there are multiple interfaces?
Let's say the sound starts in aire, then reaches a wall of glass with helium on the other side.
What do we need to know to calculate how the intensity is affected in scenarios like this one?


2 Answers 2


It sounds like you are trying to solve a problem where there are boundaries and the possibility of reflection, transmission, and absorption. As long as all the media are linear, isotropic, and homogeneous the standard exp(ikR)/R Green's function is valid in each medium. To account for the boundaries you can apply a number of computational techniques, e.g. method of images, etc. There are well known formulae for factoring in the amount of acoustic wave reflected back or absorbed by the material. I'd recommend looking at a book like Waves in Layered Media by Bekhovskikh (a classic). You still need to set up the geometry of the interfaces correctly and things may get complicated if boundaries are curved, requiring different techniques, but the basics still apply.

The net result of the interference will appear to change the intensity away form the 1/R^2 law but the basic physics that led to it is actually based on the same law you are familiar with. You would want to characterize the intensity as a function of various parameters like total distance traveled, number of layers, type of material, etc. This can be done experimentally with a controlled source, set up or materials, and a receiver moved to different locations under the same controlled conditions.

There are examples of deviations from the inverse square law that are due to propagation. Not all geometric spread is inverse square! If the medium of propagation has a continuously changing refractive index or fluid flow exp(ikR)/R will no longer be the Green's function of the Wave (Helmholtz) equation. This is known in underwater acoustics where, in the Pacific ocean, temperature and pressure gradients create a refractive wave guide independently of surface and bottom bounce. In these cases you need a more sophisticated method of describing propagation to the boundaries, e.g. ray tracing, finite element method, etc. You can still treat boundaries like you do in simpler problems but you need to accurately predict what hit the boundary.


The inverse square law is a geometrical factor causing energy radiated from a point source to lose intensity as it spreads out.

Separately there is also a reduction in transmitted intensity due to absorption within the medium in which the energy propagates. For a plane wave this follows the Beer Lambert law : transmitted intensity decays by a factor of $e^{-a x}$ where $x$ is distance travelled and $a$ is an absorption coefficient which is characteristic of the material.

When a wave such as sound or light crosses a boundary between two mediums in which there is a difference in the speed at which waves propagate, then there is also partial reflection of wave energy if the speed is slower in the 2nd medium. The amount of energy reflected also depends on the angle of incidence, generally increasing as the angle of incidence increases, except that for polarised waves the reflection vanishes at what is called the Brewster Angle.

Putting everything together, if you have a series of media and boundries for sound to pass through, you simply multiply the initial intensity by the geometrical or absorption or reflection factor for each.


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