Another way to look at it.

Take an ideal system, one which is a perfectly elastic with no damping whatsoever. In that case, the amplitude, or strength of the signal will not decay and is constant over time. The strength of the signal remains constant with time. The more quickly the system damps, the faster the decay in the displacement amplitude and the loss of the signal will be greater. Take the critically damped system as the other extreme case in which the signal is completely lost after just one cycle.

If you take two systems with the same initial displacement amplitude and frequency, the one with greater damping will result a weaker signal over time/length. Initially at T=0 (the input) they are the same, after any time=T1 the system with greater damping will have a lower amplitude.

The damping controls the rate of decay of the signal strength and there is also the frequency which dictates the rate, or period of the oscillations. When you pulse a system and let it go (as opposed to driving it) you have a complex set of overlapping frequencies. Over a given short time (length), higher frequencies will undergo many more oscillations and will be attenuated more quickly that the lower frequencies.....damping of some of the higher frequencies down the length should change the feel of the rod at the hand, somewhat like the tone or character in a musical instrument.

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