We re-examine the linear theory of wave propagation through an elastic string under uniform tension or a slender elastic rod from a perspective that focuses on the flow of mechanical energy and mechanical momentum. Continuity equations are established for the flow of energy and momentum, leading to two boundary conditions for the net wave displacement. The important special case of a small amplitude pulse of arbitrary shape traveling through a uniform slender medium joined to another medium with a different linear mass density is examined in detail. The new boundary conditions lead to the correct relative amplitudes for the reflected and transmitted pulses. We obtain the instantaneous mechanical energy and momentum of the incident, reflected, and transmitted pulses and show that the net mechanical energy and momentum are separate constants of motion. The cases of an incoming pulse described by a Lorentzian and a Gaussian distribution are suggested as problems to be solved by the interested reader.

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