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If I understand it right, the Doppler effect means that when something is coming towards you, the pitch of the sound it makes (and the light that comes from it, but that doesn't really matter for the speed I'm concerned about) gets higher as it gets closer, because the waves compress.
So why in old war movies does the whistling a bomb or a shell or whatever makes as it falls to the ground get lower in pitch the closer it gets? Is there something I'm not understanding, or is the whistling a thing that doesn't actually happen?