An application of isoperimetric inequalities to the transmission of scalar waves through small apertures

Isoperimetric inequalities bound the exact solution of complicated boundaryvalue problems through the use of certain simple geometric quantities. As an example of this concept, we propose bounds on the scalar (acoustic) transmission coeficient which characterizes a small aperture of arbitrary shape. These bounds depend solely on the area and perimeter of the aperture. The plausibility of the bounds is demonstrated through the evaluation of several known solutions. Results are noted for apertures in both flexible and rigid screens and these results are compared and contrasted with those of the corresponding vector (electromagnetic) problem.