Wave Momentum

The physical forward momentum carried by a transverse wave along a string is conveyed by a secondary longitudinal wave [394].

A less simplified wave equation which supports longitudinal wave momentum is given by [394, Eqns. 38ab]

$$\epsilon {\ddot \xi} = \left(SY+K\right) \xi^{\prime\prime} + SY\eta^\prime\eta^{\prime\prime}$$ (B.40)

$$\epsilon {\ddot \eta} = K \eta^{\prime\prime} + SY\left(\frac{3}{2}(\eta^\prime)^2\eta^{\prime\prime} +\xi^{\prime}\eta^{\prime\prime} + \eta^{\prime}\xi^{\prime\prime}\right)$$ (B.41)

$$\approx K\eta^{\prime\prime},$$ (B.42)

where $\xi$ and $\eta$ denote longitudinal and transverse displacement, respectively, and the commonly used ``dot'' and ``prime'' notation for partial derivatives has been introduced, e.g.,

$${\dot \xi} \isdef \frac{\partial \xi}{\partial t}$$ (B.43)

$${\xi^{\prime}} \isdef \frac{\partial \xi}{\partial x}.$$ (B.44)

(See also Eq.(C.1).) We see that the term $SY\eta^\prime\eta^{\prime\prime}$ in the first equation above provides a mechanism for transverse waves to ``drive'' the generation of longitudinal waves. This coupling cannot be neglected if momentum effects are desired.

Physically, the rising edge of a transverse wave generates a longitudinal displacement in the direction of wave travel that propagates ahead at a much higher speed (typically an order of magnitude faster). The falling edge of the transverse wave then cancels this forward displacement as it passes by. See [394] for further details (including computer simulations).