Some properties of the solutions of wave equations

Some properties of solutions of initial value problems and mixed initial-boundary value problems of a class of wave equations are discussed. Wave modes are defined and it is shown that for the given class of wave equations there is a one to one correspondence with the roots _i (k) or k _j () of the dispersion relation W(, k)=0. It is shown that solutions of initial value problems cannot consist of single wave modes if the initial values belong to W ₂ ¹ (–, ); generally such solutions must contain all possible modes. Similar results hold for solutions of mixed initial-boundary value problems. It is found that such solutions are stable, even if some of the singularities of the functions k _j () lie in the upper half of the plane. The implications of this result for the Kramers-Kronig relations are discussed.