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Why is the speed of sound constant in a given medium? What is the intuitive picture behind it?

Like for example, I would imagine that "hitting" the air "harder" would make sound travel faster than "hitting" it less hard?

I'm looking for a more intuitive explanation rather than a formula. I have not come across any satisfying answer to this rather simple question yet.

If I move a rope with a stroke, it feels like I would be able to control the propagating speed of the transverse wave across the rope by varying the power in the stroke?

If a medium were completely packed I could understand why the speed of sound would be constant. Here an analogy would be like filling a tube with balls. If I put in a ball in one end, another ball will fall out of the other end. The response time of the ball by the other end is the same regardless of the power of my punch by the other end. BUT. If we would add space between the balls in the tube, this would no longer be the case since I can control how fast the balls will roll between collisions.

I hope this makes some send. Any help to be able to imagine this phenomenon is very welcomed.

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  • $\begingroup$ You say you feel like it, but if you do not change the tension of the rope, are you sure that the speed of the propagating wave in the rope does depend on amplitude? $\endgroup$
    – Koschi
    May 25, 2021 at 18:21
  • $\begingroup$ I have never said anything about amplitude. Yes, I use phrasing like "feel like" because I am thinking about it intuitively. Rather the speed of the execution of the stroke. $\endgroup$ May 25, 2021 at 19:12
  • $\begingroup$ About what kind of "stroke" are we talking about here? You wrote about "power" of the stroke in your question, I am not sure what you mean by that. In ropes we typically talk about transversal waves... So we should talk about your hand's (the hand that holdes the rope) motion perpendicular to the rope to produce these waves. You can change the amplitude of this motion, changing the wave's amplitude, and you can change the speed, which does change the frequency of the wave... do you have the experience that any of these changes how fast the wave propagates through the rope? $\endgroup$
    – Koschi
    May 25, 2021 at 19:23
  • $\begingroup$ Thanks. But I am simply saying that the speed of a transverse wave by moving a rope can in principle be varied and wondering why the speed of sound waves can't be varied. $\endgroup$ May 25, 2021 at 19:29
  • $\begingroup$ Contrary to your intuition, I do not think that the speed of the wave in a rope can be varied by amplitude or frequency... since both does not change how fast the information of displacement arrives at the neighboring parts of the rope. $\endgroup$
    – Koschi
    May 25, 2021 at 19:36

1 Answer 1

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The speed of sound does indeed depend on the amplitude/loudness/excitation force. However, for sufficiently weak waves, and if we ignore a variety of additional effects (such as drag and internal friction), we can approximate the speed of sound as constant. The intuitive reason is that any smooth minimum in a curve looks like a parabola up close. I'll explain what I mean as follows.

Some background: A propagating compressive wave induces harmonic motion in the molecules of the constituent material. The key characteristics are (1) a restoring force and (2) inertia; without these, we don't have a wave. Now, if we just imagine as a simple model a weight with mass $m$ hanging from an ideal spring with spring constant $k$, then the natural frequency $\omega=\sqrt{k/m}$. Regardless of how hard we strike the weight, it'll oscillate with that same natural frequency because a larger strike increases the speed but also increases the opposing force as the mass displaces farther from its equilibrium position. The two increases scale up exactly equally in the ideal case.

A sound wave moving through a material operates in a similar way, although the nature of the restoring force varies across material classes (entropic for gases, enthalpic for condensed matter). There's a slight change from an ideal frequency of $\omega=\sqrt{k/m}$ to an ideal speed of $v=\sqrt{E/\rho}$, where $E$ is an elastic modulus and $\rho$ is the density, but the general form is the same: a stiffness term divided by an inertial term.

At this point, we have to examine the constraints of this ideal case: $k$ and $m$ (or $E$ and $\rho$) remaining constant. We don't expect the mass to vary for a uniform material moving at nonrelativistic speeds, so that takes care of inertial variation. How about stiffness variation that would tend to alter $v$?

Broadly, a material's stiffness arises from its free energy increasing as we deform it. In condensed matter, we can model this deflection–energy relationship using the pair potential, where the minimum energy corresponds to the equilibrium spacing between molecules.

The minimum energy region will generally have some type of asymmetric shape. But by Taylor series expansion, we can show that all small minima look like parabolas up close. A parabolic energy profile is the characteristic of an ideal spring with constant spring constant $k$: with increasing deformation, the restoring force scales up linearly, and the energy scales up quadratically. For this reason, most stiff solids can be characterized by a single Young's modulus for uniaxial loading, regardless of magnitude and sign of the load.

Therefore, assumptions of a constant speed of sound rely on small disturbances and an idealized simple model—much like the assumptions of symmetric linear elasticity. Alternatively, we'd say that acoustic wave dispersion is zero for small perturbations. This is usually a pretty good model, but it is a simplified one.

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  • $\begingroup$ Excellent intuitive answer. I’d like to know how this can be applied to gases, where the idea of each particle occupying a quadratic potential well isn’t obvious to me. $\endgroup$
    – dsfkgjn
    Aug 23, 2021 at 15:54
  • $\begingroup$ For gases, the stiffness arises predominately for entropic reasons (rather than enthalpic reasons, as in condensed matter), and this stiffness—namely, the isentropic bulk modulus $K_S$—depends on the pressure $P$, density $\rho$ and heat capacity ratio $\gamma$, with the speed of sound modeled as $v=\sqrt{K_s/\rho}=\sqrt{\gamma RT/M}$ for an ideal gas, where $R$ is the gas constant, $T$ is temperature, and $M$ is the molar mass. To assume nondispersion is thus to assume that the temperature and heat capacity ratio don't change much in the compression that occurs in sound propagation. $\endgroup$ Aug 23, 2021 at 18:45

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