Variational method for prediction of acoustic radiation from vibrating bodies

Integral equation formulations such as discussed by Chertock, Copley, Schenck, Meyer, Bell, Zinn, Stallybrass, and many others, typically have nonsymmetric kernels and are not easily rephrased as a variational principle. However, a symmetric kernel does result from the normal derivative of the Kirchhoff‐Helmholtz integral, with the exterior point subsequently allowed to approach the surface, but the integrand then has unmanageable singularities. A method due to Maue and Stallybrass allows this integral equation to be recast into a usable form (involving tangential derivatives of the unknown surface pressure) and this version in turn leads to a variational principle that has considerable promise for systematic approximate solutions of radiation problems in the low to moderate ka regimes. Validity of this formulation is substantiated by derivation of known results for vibrating spheres and disks. Numerical results for other geometries are expected to yield higher accuracy than previously because one can incorporate one's best “physical insight” in initial choices for the class of trial functions for surface pressure. If computations now in progress are completed, the example of an oscillating airfoil will be discussed, with some expected clarification of the anomalous results of the Brooks' experiment.Integral equation formulations such as discussed by Chertock, Copley, Schenck, Meyer, Bell, Zinn, Stallybrass, and many others, typically have nonsymmetric kernels and are not easily rephrased as a variational principle. However, a symmetric kernel does result from the normal derivative of the Kirchhoff‐Helmholtz integral, with the exterior point subsequently allowed to approach the surface, but the integrand then has unmanageable singularities. A method due to Maue and Stallybrass allows this integral equation to be recast into a usable form (involving tangential derivatives of the unknown surface pressure) and this version in turn leads to a variational principle that has considerable promise for systematic approximate solutions of radiation problems in the low to moderate ka regimes. Validity of this formulation is substantiated by derivation of known results for vibrating spheres and disks. Numerical results for other geometries are expected to yield higher accuracy than previously because one can inc...