### Wave Momentum

The physical forward*momentum*carried by a transverse wave along a string is conveyed by a secondary

*longitudinal wave*[391].

A less simplified wave equation which supports longitudinal wave momentum is given by [391, Eqns. 38ab]

(B.40) | |||

(B.41) | |||

(B.42) |

where and denote longitudinal and transverse displacement, respectively, and the commonly used ``dot'' and ``prime'' notation for partial derivatives has been introduced,

*e.g.*,

(B.43) | |||

(B.44) |

(See also Eq.(C.1).) We see that the term 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 [391] for further details (including computer simulations).

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Non-Stiff String