The physical forward momentum carried by a transverse wave along a string is conveyed by a secondary longitudinal wave .
A less simplified wave equation which
supports longitudinal wave momentum is given by [394, Eqns. 38ab]
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).