The Sommerfeld Problem revisited: Solution spaces and the edge conditions

AbstractThe Sommerfeld Problem is reexamined with a view to obtaining rigorously and with sufficient generality its differential and integral formulations, with emphasis on the latter. The authors derive a mathematical condition for the energy stored in the field to be finite and show that this condition is equivalent to specifying the space of solutions to the classical Wiener-Hopf equation of half-plane diffraction (the Sobolev space H−12, 2+(R)). This equation is then solved in H−12, 2+(R) for any excitation in H12, 2(R+). The behaviour of the solution near the edge is discussed for a class of excitation functions