The physics S.E. may be wrong for this question, but I wonder what happens to a vibration in broken, cracked, or otherwise damaged cylinder. Say you have a 1 meter long steel cylinder, you break it in two, and then put the pieces back together (without welding or anything). You then put in a sinusoidal vibration in one end, and measure the frequency and amplitude in the other end. As far as I can tell, the frequency would not change, but only the amplitude, as energy of the frequency wave is lost in the translation from the complete steel cylinder to the damaged cylinder and back to the complete cylinder again. Is this interpretation correct? If so, how would a cylinder that is only cracked or bent behave? I know they use vibration to, for example, diagnose motors and propellers.
If you send a longitudinal wave through a broken pipe, then you can transmit the positive pressure part of the wave without problems, but when you reach the tension part of the wave, it is not transmitted but reflected. This leads to a complex wave pattern - the transmitted wave will be "clipped", resulting in harmonics. This is why a cracked bell, tube, etc, sounds "clunky" - instead of a nice pure note, you hear these higher harmonics.
For smaller cracks, there will be partial reflection but the principle is the same.
It will almost certainly change the resonant frequency for longitudinal waves - except for very specific locations of the discontinuity.