On the conservation of momentum for a sound pulse reflecting from a pressure-release boundary

When a "massless" one-dimensional sound pulse (mass of a sound pulse is defined as an integral of the perturbation of density over the pulse length) reflects from a pressure-release boundary, its momentum changes sign. This obviously violates momentum conservation. However, in contrast to the case of an unbounded medium, calculation of the momentum in a bounded region includes a second-order term as well. Apparently, the second-order correction to the linear solution ensures momentum conservation in this case. The purpose of this Letter is to find a concrete form of this second-order correction. It appears that, as a result of the nonlinear interaction of the pulse with a pressure-release boundary, the latter experiences second-order net shift. This leads to the generation of a massive second-order rarefaction pulse whose momentum is directed opposite to the direction of propagation of the pulse itself. Appearance of this pulse ensures total momentum conservation.