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

It is shown that the wave equation ψ_xx-ψ_yy=0 for the field ψ(x,y) in the domain R(xy) can be transformed into a wave equation Ψ_ξξ-Ψ_ηη=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 ξ_x= ±η_y and ξ_y=±η_x "±" 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).