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If a car was going at near mach-1 speed heading towards a wall and honked its horn. What frequency would the driver measure the reflected sound wave? I trying to think about this reasonably, but the answer I get is somewhat not what I initially expected.

Here is my current thinking, correct anything you might deem wrong. First let's pretend there is no reflection but instead the wave comes from another source moving head on to the observer with equal and opposite velocity. So then we have the receiver moving towards the source, and the source moving towards the receiver.

Using the Doppler shift equation, the observed frequency is upshifted by a factor of (c+v)/(c-v), where c is the speed of the sound wave, and v the speed of the car. I suppose the interesting thing is that the source and receiver are actually the same, but there is still a shift due to the reflection. Or, maybe I made a mistake in my logic.

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I think you are exactly right. In fact the method you used of imagining a second moving car is exactly what you would do if you were trying to find the pressure as a function of position in the original problem with the wall. The technique is called the method of images.

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I usually work this as two sequential Doppler shift problems. First, treat the wall as the detector. Then let the wall become the source of a wave with the frequency it detected and let the driver become the detector.

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