We investigate the effect of boundary conditions on the low temperature phonon transport in a quantum waveguide with structural defects by using the scattering-matrix method. A comparison between transmission coefficients is made when Neumann and Dirichlet boundary conditions are applied for acoustic modes, respectively. The result indicates that the behavior of the transmission coefficient versus reduced phonon frequency is qualitatively different for different boundary conditions. The stop-frequency gap and the transmission resonance are observed for certain structural parameters due to mode-mode coupling.