Membrane Wave Explorer

Live Script exploring waves on membranes with various shapes, isotropic and anisotropic stress distributions, and forcing conditions.

This educational Live Script explores equilibrium stress and flexural waves on flat isotropic elastic membranes with various shapes, stress fields, mass density distributions and other conditions. The script introduces the theory, illustrates how finite element analysis with the PDE Toolbox can solve the hyperbolic wave equation in circumstances not solvable by analytic means, and explores a variety of wave phenomena.

The illustrative default application is to ribbon-shaped membrane with circular cutouts in the sides that create two weakly linked regions and with an impressed anisotropic stress distribution. The script first finds and visualizes the normal modes and their frequency spectrum. The results illustrate mode shape morphing and frequency shifts due to shear and the weak link. It then launches a Gaussian pulse while shaking an edge harmonically with damping creates an animation illustrating wave transmission and reflection at the weak link location, the decay of the transient, and the development of a steady state response.

A theoretical introduction clarifies the physical assumptions behind the membrane wave equation. It shows how equilibrium stress fields may be derived from multinomial solutions to the Airy equation and checked to avoid wrinkling instability. The corresponding external stress on the boundaries may be derived from these solutions and visualizations of the strain field and boundary stress are provided. Plane wave solutions are shown to generally have direction-dependent local phase velocity with acollinear group velocity.

The script shows how to generalize the geometry to an arbitrary shape and multiple regions. Provided stair-step and linear mass density variation options allow studies of, for example, reflection and transmission at interfaces between regions of different wave speed and of the membrane version of the hanging chain or whip and 2d acoustic black holes. The examples may be generalized to arbitrary mass density variations such as a distribution of inhomogeneities.

The boundaries may be fixed, float up and down out of plane, or satisfy mixed Neumann boundary conditions. One illustrative edge may be shaken up and down following a harmonic function of time governed by amplitude and frequency parameters to explore resonances, diffraction, and scattering. This method of forcing is easily generalized to arbitrary functions and boundaries. A Rayleigh damping model may be enabled to study damping-associated dispersion and it is shown how to customize damping at the node level. A static or dynamical external force density of any shape may be applied to study the equilibrium shape under inflation and states excited by acoustic pressure or a drumstick.

With this script, one may explore and visualize general wave phenomena including scattering, tunneling, impedance matching, dispersion, resonance, diffraction and specialized applications like determining the normal mode frequencies of a drum head with arbitrary shape, nonuniform stress, or inhomogeneities.

Membrane Wave Explorer may interest students and instructors in physics and related fields as well as musicians and the curious. The theory is accessible to enthusiastic students who have encountered waves and elementary elasticity theory and who are familiar with multi-variable calculus, matrix algebra, and differential equations. The steps to create simulations with the PDE Toolbox are carefully explicated and might serve as an introduction to its application to 3D problems. 'Try this' interactive suggestions, more involved 'Challenges', and many references and hyperlinks are included for further exploration. The less involved 'Challenges' illustrate the range of experiments possible without additional coding. A number of functions are embedded in the script so it is self-contained but these may be extracted and saved and the background information removed to create a more concise operational Live Script.