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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]

$\displaystyle \epsilon {\ddot \xi}$ $\displaystyle =$ $\displaystyle \left(SY+K\right) \xi^{\prime\prime} +
SY\eta^\prime\eta^{\prime\prime}$ (B.40)
$\displaystyle \epsilon {\ddot \eta}$ $\displaystyle =$ $\displaystyle K \eta^{\prime\prime} +
+\xi^{\prime}\eta^{\prime\prime} + \eta^{\prime}\xi^{\prime\prime}\right)$ (B.41)
  $\displaystyle \approx$ $\displaystyle K\eta^{\prime\prime},$ (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.,
$\displaystyle {\dot \xi}$ $\displaystyle \isdef$ $\displaystyle \frac{\partial \xi}{\partial t}$ (B.43)
$\displaystyle {\xi^{\prime}}$ $\displaystyle \isdef$ $\displaystyle \frac{\partial \xi}{\partial x}.$ (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).

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University