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In the Doppler effect, we see that the frequency of sound that the observer hears changes according to the motion of the observer and source. For example, if the source is moving towards the observer, who is stationary, the frequency appears to increase. But if the frequency increases, shouldn't the energy associated with the wave also increase? Where does this energy come from?

According to the sound energy equation, the energy varies with the square of the velocity of the particles ( https://en.m.wikipedia.org/wiki/Sound_energy ). But doesn't that mean that the energy will increase no matter what direction the the particle is moving in?

Isn't this contradictory to the frequency given by Doppler's equation, which says frequency(and hence te energy) decreases if the source is moving away from the observer.

As a possible solution to this, can we consider the Doppler effect to be caused not by frequency changes, but by changes in perceived intensity of sound because of distance between source and observer?

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To see how the Doppler effect is consistent with energy conservation, consider a source that emits a power $P$: In one second, it sends $P$ joules toward the source.

At the start of that second, it's a distance $d$ away. $d/c$ seconds later, that energy $P$ starts to arrive at the target. But the source is moving toward the target at velocity $v$. That means after one second, it's closer by a distance $1 \times v$, so the end of that energy $P$ takes only $1-v/c$ seconds to get there.

So the source seems to have intensity $P \over (1 - v/c)$ to the target, which is exactly the Doppler shift.

Bottom line: Each cycle of the wave carries the same energy at is always did, and since the Doppler blue-shift means more of those arrive per second, everything works out.

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if the frequency increases, shouldn't the energy associated with the wave also increase?

It should.

Where does this energy come from?

It must be coming from a moving object: transmitter, receiver or both. The extra work is presumably performed by a transmitting or a receiving element pushing the air in front of it.

According to the sound energy equation, the energy varies with the square of the velocity of the particles ( https://en.m.wikipedia.org/wiki/Sound_energy ). But doesn't that mean that the energy will increase no matter what direction the the particle is moving in?

It does not. The average speed and average kinetic energy of an oscillating air molecule, measured by the receiver, will be higher if the transmitter moves toward the receiver and lower, if the transmitter moves away from the receiver.

As a possible solution to this, can we consider the Doppler effect to be caused not by frequency changes, but by changes in perceived intensity of sound because of distance between source and observer?

The essence of the Doppler effect is frequency change - not the change in the perceived intensity - and that frequency change is easy to verify.

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