Acoustics: Sound Fields, Transducers and Vibration, 2nd Edition

14.6. Membrane wave equation in polar coordinates

As the Laplace operator of Eq. (14.5) is the same as that of the wave equation in three dimensions given by Eq. (2.147) but without the z-term, it follows that the Laplace operator for the membrane in polar coordinates, which is given by

∇ 2 = ∂ 2 ∂ w 2 + 1 w ∂ ∂ w + 1 w 2 ∂ 2 ∂ ϕ 2 .

(14.25)

A rigorous derivation of this is given in Ref. [1]. Following the same procedure as in Section 2.10, where we separate the wave equation into its radial and angular (azimuthal) parts, we arrive at the solution to the homogeneous membrane wave equation