where and denote longitudinal and transverse displacement, respectively, and the commonly used ``dot'' and ``prime'' notation for partial derivatives has been introduced, e.g.,
(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  for further details (including computer simulations).
Particle Velocity of a Gas