In this chapter and the next, we will derive results that were used in previous chapters to study transducers and their radiation characteristics. The aim is to provide insight into how the shape of a transducer determines its behavior as well as an understanding of how to solve acoustical problems analytically. In each problem, a new concept or method is introduced so that each problem is slightly more complicated than the previous one. Formulas are given which the interested reader may use as part of his or her own simulations. In this chapter, we will take the wave equation solutions of
Chapter 2 and apply the appropriate boundary conditions to them to determine the unknown coefficients. This is known as the boundary value method. In fact, we have already used this method to solve for the reflection of a plane wave from a plane in
Section 4.9, radiation from a pulsating sphere in
Section 4.10, and an oscillating sphere
Section 4.15. In
Chapter 13, we will treat sound sources as arrays of point sources, which are integrated using the boundary integral method.