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Say an airplane flies at Mach 1. The doppler effect causes the front of the sound emitted by the plane to reach infinite frequency:

$f = \left ( \frac {343+0}{343-343} \right ) f_0 = ∞$

If the plane suddenly stopped, and there was an observer in front of the sound waves to receive the sound, what would the bunched up waves sound like to the observer? Would they cancel each other out and cause no sound?

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    $\begingroup$ Have you heard of the sonic boom? $\endgroup$ – tpg2114 May 17 '16 at 1:03
  • $\begingroup$ Some nice examples of shock waves due to supersonic flight youtube.com/watch?v=gWGLAAYdbbc $\endgroup$ – Farcher May 17 '16 at 14:11
  • $\begingroup$ @tpg2114 Yeah I heard of it and so what... I studied aerodynamics and acoustics and the problem is much more complicated than what usually people explain and commonly understand it... so instead of just typing "Have you heard of the sonic boom? " why don't you give a more complete answer $\endgroup$ – Federico Gentile Apr 25 '18 at 9:33
  • $\begingroup$ @FedericoGentile The user described exactly what causes a sonic boom -- by asking if he/she heard of it, I was trying to understand what background the user is coming from. It's not my fault that the user never followed up to say yes or no. If they had said yes, then I would have followed up with what they didn't understand about a sonic boom, so I could write a good answer to address the actual issues. If they said no, then I would write that up as an answer. Since they said nothing, I didn't return to the question until 2 years later, when you decide to criticize me by making assumptions. $\endgroup$ – tpg2114 Apr 25 '18 at 14:11
  • $\begingroup$ @tpg2114 I will criticize every time i see sarcasm... especially on this platform where there is an excess of it $\endgroup$ – Federico Gentile Apr 26 '18 at 7:10
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There is no such thing as "infinite hertz". The equation to calculate the doppler shifted frequency breaks down when the emitter travels at, or faster than, the speed of sound. What actually happens when an object travels that fast is the sound waves all pile together and form a shock wave. This is what creates the sonic boom.

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First off, its important to remember that such models are just models. Most of the wave propagation equations you are using assume a linear transfer medium... aka the atmosphere.

When you get into extreme situations like your infinite frequency, those assumptions start to break down entirely. They cease to be good models of how the world works. As such, you need to re-define the problem in terms which are better captured.

Take the sonic boom as an example. We assume "pressure information cannot propagate faster than the speed of sound in the medium." We also assume that all objects have a "boundary" layer of gas particles with a relative velocity of zero with respect to the object. Put them together, along with an object traveling faster than mach 1, and something has to give. You can't achieve all of those goals simultaneously. What ends up happening? We get a "shock wave" that's a few nanometers thick where we can't use our nice easy wave-mechanics models of the atmosphere. We have to treat the molecules in the air as individual particles in all their glory, and all sorts of funny effects come up. Everything behaves reasonably on both sides of this boundary, but they behave so poorly in those few nanometers that we typically just treat it as a discontinuous boundary and solve the equations on both sides.

So, recognizing that the models will almost certainly break down, what happens in your infinite frequency case? The first question is how do we define the wave you want to look at? We don't have enough information. Frequency is not enough. You also need amplitude or energy if you want to talk about what something will sound like.

So what if you hold amplitude constant and increase the frequency to infinity? Well, the answer is that you tear the universe apart. The power of a wave is proportional to both the square of the amplitude and the square of the frequency: $P\propto (Af)^2$. If you just increase the frequency towards infintiy while holding A constant, your power increases unbounded. If this wave is any larger than a single point, that means unbounded energy densities. You start to get into relatavistic effects and end up either swallowing up the universe in a black hole or otherwise ripping apart space-time as we know it! Let's not do that.

So what happens if you hold energy constant, to avoid this relatavistic nightmare? Well, to do this while frequency goes up, amplitude must go down. At infinite frequency, the amplitude must be zero, so the particles in the medium are not being moved any meaningful distance.

This will clearly violate many of the assumptions of the wave equation. It's going to be hard to argue that the molecules bashing together at a huge-but-finite rate are going to meaningfully propagate an infinite frequency wave. Realistically speaking, gases will exhibit a high frequency roll-off in this respect. They will attenuate waves as they get to higher frequencies.

As a result, your infinite frequency wave, with some reasonable assumptions to fill in the missing pieces, will sound like nothing at all. It will not transmit through the air meaningfully, because the medium attenuates such frequencies.

And even if it did get the sound to your ear, your ear also attenuates high frequencies. It has a bandpass like behavior built into it that prevents us from hearing sounds higher than roughly 22kHz.

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Tpg2114 is right. It's the sonic boom. All the wavefronts in front of the airplane bunch up and add up, and you hear a thump. It travels with the aircraft. To go past it is like going through a barrier because of the high pressure there. Once you pass it the sonic boom trails behind the aircraft.

See it at http://www.acs.psu.edu/drussell/Demos/doppler/doppler.html

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And actually at infinite Hertz, the air will not vibrate, so we'll hear nothing. And even if it does, our hearing ability is only till 20kHz. It'll be like passing direct current through a speaker, the membrane is vibrating, but unfortunately we hear nothing!

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