Why do waves always travel fastest in the least dense medium (Boundary Behaviours)?

Question

Why do waves always travel fastest in the least dense medium? I thought waves are vibrations and wouldn't this means that waves travel faster in a denser medium since they are tightly packed which allows less time for molecules to bump into each other? With that said, why does the wavelength is greatest in the least dense medium?

Answer 1

Wave propagation is due to one "chunk" of a material exerting a force on a neighboring chunk, which then exerts a force on its neighboring chunk, and so forth on down the line. The speed at which a wave propagates is then really down to two factors: if I move one "chunk" of the material, how quickly does its neighboring chunk respond?

This question can really be answered just by thinking about Newton's Second Law, $F = ma$. A "chunk" of matter will accelerate faster (and thus the wave will travel faster) if either:

• Its neighboring chunk exerts more force on it (i.e., $F$ is bigger), or
• It has a smaller mass (i.e., $m$ is smaller).

The first factor is what we experience as the "stiffness" of the material: how easily does it deform to a particular applied force. For a wave on a string, this is determined by the tension in the string. For waves traveling in a 3D medium, this is determined by various elastic moduli of the medium, such as Young's modulus or the bulk modulus. All other things being equal, a stiffer material (such as steel) will have a higher speed of wave propagation.

The second factor is what you're thinking of: if each "chunk" has a smaller mass, then it can respond more quickly to the motion of its neighbors, and so the wave travels faster. All other things being equal, a less dense material will have a greater speed of wave propagation. In the common classroom example of two strings of different densities that are tied together, the "stiffness" of the media is determined by the tension in the string, which is the same in both media; and so the waves travel faster in the lighter string.

Note, however, that it's entirely possible for waves to travel faster in denser materials, so long as they're sufficiently stiff. As pointed out by @MarkH in the comments, sound travels much faster in most solids (dense but very stiff) than it does in air (not dense but not very stiff at all.) So be sure to think carefully when you extrapolating this classroom demonstration to real-life behavior.