I have two questions about the physics of sound. As a background, I know the process of sound production can be understood as 3 stages that happen continuously:

  1. An object oscillates back and forth and displaces the molecules of the medium. The result is a change of density (When the object oscillates to the front, it also pushes the nearest molecules to the front, increasing the density in a certain region. When it oscillates to the back leaving a "vacuum", there is a decrease of density)

  2. The change of density is related to a change of pressure (regions with higher density of molecules will have higher pressure)

  3. The change of pressure leads to the displacement of molecules (movements from higher pressure to lower pressure regions)

My first doubt is: Is there some kind of threshold imposed to the beginning of the object vibration that results in the production of a wave travelling at the speed of sound? Since every movement leads to a displacement of the air, there must be a condition that separates the phenomena of sound and simply the dragging of the air.

My second doubt is: When an object starts its vibration, it moves forward, thus "pushing" the molecules forward, and then backwards when it leaves a region with a lack of matter. The molecules that were ahead come back to fill the "vacuum" due to pressure difference. In their way ahead, they push yet another column of molecules, and so a wave of molecules being pushed and going back to fill a gap is formed. Why is the case that the molecules that were ahead go back to fill the void and not the molecules that were behind? If that were the case, there would also be a dragging of the air.


Doubt 1: Feynman gives some information in its lectures:

" (...) if an object is moved at one place in the air, we observe that there is a disturbance which travels through the air. (...) Of course, if the object is moved gently, the air merely flows around it, but what we are concerned with is a rapid motion, so that there is not sufficient time for such a flow."

The threshold is about a "rapid motion". Since the velocity of the object will be completely determined by the amplitude and frequency of the movement, what is exactly this speed he talks about? Is he really talking about this amplitude/frequency combination (when the object is vibrating and there's not only one single push)? For example, if the amplitude is high, the frequency couldn't be too low or the average speed is too low.

  • $\begingroup$ An interesting side note: when sound (energy propagated by compression an rarefaction of pressures) waves move through fluids in motion (wind, current, flow etc.) there will be a Doppler shift in the frequency of the sound relative to an observer at rest. The size of the shift can be used to measure the flow rate, wind speed, etc. Commonly done with ultrasonic sensors. $\endgroup$ – docscience Jan 4 '17 at 21:32

We see sound waves in regions where it is reasonable to treat the air as an elastic medium. The speed of propagation in this medium is defined by $\sqrt{\frac{K_s}{\rho}}$, where $\rho$ is the density and $K_s$ is the coefficient of stiffness. For gasses, this coefficient is the bulk modulus which is the ratio of the infinitesimal change in pressure due to a relative decrease in volume ($K_s=-V\frac{dP}{dV}$).

As long as air behaves elastically, we can use these equations and others to derive the behavior of vibrating waves in the air. This model falls apart, however, when we leave the region where it behaves elastically. This occurs when you start traveling fast enough (naturally, around the speed of sound is the threshold).

Let's think about our loudspeaker. The surface of the speaker does not actually travel at the speed of sound. It travels at a lesser velocity defined by the amplitude and frequency of the sound. The position of the speaker is $x=A\cos\omega t$, so the velocity is $v=A\omega\sin\omega t$, this means the maximum velocity of the speaker is $A\omega$. This is also the maximum velocity of any particle of air displaced by the wave. The wave itself moves at the speed of sound, but the individual particles move at a maximum of $A\omega$.

As long as $A\omega$ is small, it is very reasonable to model air as an elastic medium, so we see normal wave propagation. However, as your amplitude or frequency (or product of the two) increases, so does the velocity of the loudspeaker, and the velocity of any air that makes up the soundwave. As $A\omega$ reaches the speed of sound, it is no longer reasonable to assume that air is elastic. $-V\frac{dP}{dV}$ ceases to be a constant, and we start to see non-linear effects. These non-linear effects are colloquially known as "the sound barrier." This is the threshold you are intuiting, and it is clearly dependent on both amplitude and velocity.

Staying below this effect, it's now safe to use the elastic equations to describe transverse waves. Now we can answer your question about what differentiates sound waves from wind. Both are pressure effects, for sure, but sound waves are a special case where the source of the pressure is well modeled as a cyclic source. Impulse sounds, such as a gunshot, straddle this line. They can be seen either as a wide-spectrum sound wave, or as a very sharp pressure wind. We use the former image when talking about what we hear, and the former for showing what it does to push the bullet forward.

We can also address your question about why air doesn't fill in backwards. If we can assume an elastic medium, we know that the position and velocity of any particle movements are a function of cosine and sine respectively. With this, we can see that the reason the air only fills in from one side is that the air on the other side is in the process of moving away from that vacuum at that point in time. In fact, that air gets slowed down by this vacuum to a standstill, and that's what shapes the next cycle of the wave!

  • $\begingroup$ Well, the threshold I was intuiting was the lowest one, not a highest. But your last paragraphs and Shashaank's comment answer my questions. $\endgroup$ – Rick Jan 22 '17 at 3:04

Your point 1 is correct and the answer is based on that. Yes , there is a threshold imposed in the beginning. Sound doesn't start off without something vibrating. There has to be source of vibration like the membrane of a drum. It vibrates and in turn puts the surrounding air into vibration. For the second doubt , you yourself did say that sound is due to oscillations. The molecules vibrate about their initial position. When they are hit by the neighbouring ones they go one way and due to inertia as well as elasticity (intermolecular forces) return back to their initial position going the other way. Sound cannot be generated in a medium which doesn't have inertia or elasticity (vacuum). So sound (wave) is not wind because there is never any net displacement of matter , just an oscillation.

  • $\begingroup$ Ok, but then my doubt remains. What exactly is the threshold? Is it related only to some initial speed of the object or the amplitude/frequency combination? (I edited the post) $\endgroup$ – Rick Jan 10 '17 at 0:26
  • 1
    $\begingroup$ @Rick According to me if you can oscillate an object in air , it will give the oscillation to the air. Doesn't the membrane of drum do the same. Off course we need a rapid movement of the object (atleast above 20Hz to hear) . If the object is moved very slowly the air sort of receives a translatory hit but we pull the object very quickly the air wouldn't get time to flow away due a sudden drop in pressure (rarefaction) the molecules will oscillate back. And different people have different thresholds for hearing too $\endgroup$ – Shashaank Jan 10 '17 at 6:45
  • $\begingroup$ @Rick Like you said a large amplitude but very less frequency wouldn't give a visible wave. Rapidity is off course needed. The amplitude needed to hear depends on the person $\endgroup$ – Shashaank Jan 10 '17 at 6:49
  • $\begingroup$ The comment before the last one is enlightening. I wish I could accept two answers. $\endgroup$ – Rick Jan 22 '17 at 3:04

Your question is not entirely clear to me, in particular what you mean by "just wind". Sound is a pressure wave and your points 1, 2 and 3 are consistent with each other and all true.

What's important to keep in mind is that sound is described with the referential of what humans can perceive and interpret and thus the boundaries in amplitude and frequency (i.e. what "ultra"- or "infra"-sound mean) are arbitrary.

As to the threshold you mention, there are a few different aspects to consider, one is linked to the viscosity of the medium. What prevents very small amplitudes to propagate is a very quick attenuation of the wave. This is described (for most practical frequencies) by Stokes' law of sound attenuation:

$$ A(d)=A_0e^{-\alpha d} $$

where $A(d)$ is the amplitude at distance $d$ and $\alpha$ is defined as: $$ \alpha=\frac{2\eta\omega^2}{3\rho V^3} $$

where $\eta$ is the dynamic viscosity coefficient of the fluid, $\omega$ is the sound's frequency (or pulsation), $\rho$ is the fluid density, and $V$ is the speed of sound in the medium. So sound with very small amplitude disappear very quickly.

As for your second point, you need to either consider a 2 dimensions simplification or an ideal planar wave where there is no other molecules to fill the gap other than the ones that were initially displaced. Or you can consider a spherical wave, where again, if the pressure increases in a spherical region uniformly, the same molecules will fill the gap left by the withdrawal of the membrane.

  • $\begingroup$ According to your answer, the threshold I am looking for is in the amplitude. Waves with small amplitude dissipate faster. However it is still an attenuation of a wave travelling at the speed of sound, right? My question is what makes a movement turn into a wave travelling at that speed. I could start a vibration with high amplitude (oscillating my hands in the air) that doesn't generate a pressure wave. $\endgroup$ – Rick Jan 10 '17 at 0:14

going to option 1 is correct. and yes the threshold is imposed to the beginning. and as in the case of waves rather it be sound or also during the conduction of heat in metal, it is not the molecule that moves forward but the vibration of molecule in their initial position and the vibration is moves forward.


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