Linear acoustics

Abstract

The wave equation may be derived from the nonlinear system of Navier—Stokes equations for small vibrations. In the present chapter we want to treat the linearized Navier—Stokes equations, a first-order system, directly. This approach has the advantage of simply leading to a self-adjoint operator and to solutions with finite energy. Furthermore the decomposition into incoming and outgoing waves is within easy reach. The disadvantage on the other hand is that the dimension of the null space of the underlying operator A is infinite; this complicates some of the proofs. The same problem arises when dealing with Maxwell’s equations in the next chapter. Therefore in Chapter 9 we shall present a unified approach to both systems, suggested by R. Picard (1985), which avoids this difficulty and shows that the system of linear acoustics and Maxwell’s equations mutually ‘elliptize’.