A simple problem for the scalar wave equation admitting surface-wave and AH-wave solutions (Q1821273)

From the authors' introductory paragraph: ''It has recently been found that the surface- and AH-wave solutions of a classical problem for Maxwell's equations are generated by slutions of a simple problem for the scalar wave equation in \(R^ 3\), namely (1) \[ (\partial^ 2_ t-c^ 2_ 0 \Delta)\phi (x,t)=0,\quad x_ 3>0,\quad t>0;\quad (\partial^ 2_ t+k\partial_ t-c^ 2 \Delta)\phi(x,t)=0,\quad x_ 3<0,\quad t>0 \] \[ \phi (x,0^+)=f(x),\quad \partial_ t\phi (x,0^+)=F(x);\quad c^ 2 \partial_ 3\phi (x',0^-,t)=c^ 2_ 0 \partial_ 3\phi(x',0^+,t) \] \[ \partial_ t\phi (x',0^-,t)+k\phi(x',0^- ,t)=\partial_ t\phi (x',0^+,t) \] where \(c_ 0>c>0\) and \(k>0\) are constants and \(x'=(x_ 1,x_ 2)\). In the present note we show that problem (1) is uniquely solvable for a certain class of initial data (f,F) and present the explicit form of the surface- and AH-wave solutions to (1). We further show how to construct the corresponding solutions to the classical problem for Maxwell's equation (2) from the solutions of (1)...''.