Formulation of Wave Propagation in Infinite Media by Normal Coordinates with an Application to Diffraction

In the theories of acoustical and electromagnetic vibrations of enclosures, as well as in field theory and electrodynamics, one quite commonly uses normal modes as generalized coordinates in Hilbert space. Here the method is extended to unlimited or partially limited mechanical media, essentially by first solving the problem for an enclosure and going to a limit while expanding all or some of the boundaries to infinity. This leads to a very useful technique, of somewhat more generality than analogous procedures used in field theory. Thus it is applicable to all non-dissipative mechanical continua, for any boundary conditions, irrespective of the order or number of differential operators describing the continua and regardless of whether the coordinate systems are separable or not. A general orthonormality condition valid in all such cases together with the necessary rules for dealing with divergent normalizing coefficients are easily obtained by limiting procedures. An equally general formulation for arbitrary sources is obtained from the principle of virtual work. The method is illustrated by simple examples, for the case of a point source representing the instantaneous injection of a unit volume, i.e., an idealized, infinitely rapid, explosion. It is shown that in the problem of diffraction by a rigid wedge or corner one is led very quickly to an explicit solution in closed form, involving elementary functions only. Some physical implications of this solution are mentioned briefly. The advantages of the normal coordinates method are discussed when thus used in propagation or diffraction problems in unlimited or partially limited mechanical continua. Obvious advantages are its generality, its flexibility in dealing with arbitrary sources, and the fact that it leads directly to the progressive transient solution.