Equal-phase two-wave membrane modes ($k_y=(2j+1)\pi /a,k_z^2=(\omega /c{)}^2-k_y^2$)

Two equal complex amplitudes with opposite transverse wavevector components sum to a cosine across a strip with edges $y=\pm a/2$. Identical phases satisfy the edge condition only for the odd transverse indices displayed. Equal magnitudes with opposite phase give the sine family; together the families form all Dirichlet strip modes $n\pi /a$, $n\ge 1$. The lowest axial phase velocity exceeds the unrestricted membrane wave speed, while its group velocity is smaller; neither should be substituted for the other.

Below the lowest strip cutoff, the axial wavevector is imaginary. A localized harmonic drive produces a displacement decaying as $e^{-\kappa |z-z_0|}$ away from the source in the dominant lowest mode. Cross-sections still oscillate in time, but there is no far-field travelling axial wave. Several more rapidly decaying transverse modes can contribute near the source. For a symmetric central point drive, transverse modes vanishing at the centre are not excited.