Covariant transformations of wave equations for initial-boundary-value problems with moving boundary conditions

Abstract

It is shown that the wave equation ψxx−ψyy=0 for the field ψ(x,y) in the domain R(xy) can be transformed into a wave equation Ψξξ−Ψηη=0 for the field Ψ(ξ,η) in the domain S(ξη). The transformation is accomplished through a complex function F(x,y)=ξ(x,y) +iη(x,y), which is not analytic. For the transformation to exist, the real transformation functions ξ=ξ(x,y) and η=η(x,y) have to satisfy wave equations in the domain R(xy) and the first-order partial equations ξx=±ηy and ξy=±ηx [‘‘±’’ distinguishes transformations of the first (+) and second (−) kinds]. Thus, the hyperbolic transformation theory is different from the conformal mapping theory, where the real transformation functions satisfy the Laplace equation and the Cauchy–Riemann conditions. As applications, the linear Lorentz transformation and nonlinear mappings of time-varying regions into fixed domains are discussed as solutions of the indicated partial differential equations. Furthermore, an initial-boundary-value problem for the electromagnetic wave equation with moving boundary condition is solved analytically (compression of microwaves in an imploding resonator cavity).