Why do mechanical waves reflect at a boundary?

Question

I am trying to better understand why mechanical waves, primarily longitudinal acoustic waves, reflect at a boundary.

I am currently understanding that, for all waves, variation in media characteristics across a boundary is what modulates wave reflection. In the case of acoustic waves, a difference in impedance between adjacent media (impedance mismatch) is what determines the extent of wave reflection and transmission. As an example, say that the impedance for two adjacent media is $Z_A$ and $Z_B$, where $Z_A < Z_B$. This indicates that for a given driving force, the amplitude of the wave in media A will be greater than that of media B. Thereby, as an acoustic wave propagates from media A into B, the amplitude (and thereby energy, as frequency is constant) will decrease. Energy conservation laws tell us that this energy must have gone somewhere. Because our measure of energy in this scenario is wave amplitude, and it has been lowered in media B, some energy must have been left behind in media A. It is this 'lost' energy that is utilized in the production of the reflected wave.

I am wondering if it is at all possible to use a microscopic description, rather than an energy-based argument, to describe why wave reflection occurs. Specifically, would it be possible to describe wave reflection using coupled harmonic oscillators or particles?

Answer 1

Certainly it is. to accomplish this, we simplify the problem to one dimension (a long, stretched coil spring is ideal) and then place a discontinuity at a point along the length of the spring. then we perturb the spring at one end and make movies of what happens when the wave on the spring travels down its length and strikes the discontinuity.

The two kinds of discontinuity are 1) attaching a weight to the spring and 2) coupling the spring to a second spring with less mass per unit length than the rest of the spring. Both of these represent impedance mismatches, and the results can be described in words, as follows.

taking the case of the weight clamped onto the spring: when the wave hits the weight, it finds it cannot easily deflect the spring where the weight is because of the inertia concentrated at the weight attachment point. In the limit of infinite mass, the weight does not move at all and all the wave energy bounces off it and travels backwards. Since the reflected wave did not move the mass, there is a deflection minimum at the reflection point and the wave snaps into an upside-down version of its original self (180 degree phase shift) and the reflected wave is a phase-inverted version of the original incident wave.

For the case where the discontinuity represents missing mass, the incident wave finds it very easy to deflect the discontinuity point upon striking it, and the discontinuity point then becomes a deflection maximum. Since the incident wave moved the discontinuity point significantly, when the deflection point relaxes back to its undeflected state the reflected wave is a mirror-image of the incoming wave (no phase shift).