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

Part XXXX: Shells

14.16. Shell wave equation in polar coordinates

Like a plate, a shell has bending stiffness, but it has additional stiffness because of its curvature. If you hold a sheet of paper by the edge, it will hang down because it has no stiffness. However, if you curve it even slightly, it will support small objects such as a pencil. The dynamic shell wave equations [9] are obtained by adding an axial inertia term to the static shell equations [8]. We shall simplify our analysis by assuming that the shell is shallow, or the radius of curvature R is large, so that we can ignore radial and tangential components of the displacement η ˜ . This is a reasonable assumption for height/radius ratios up to around 0.25. The following ...