SOME PROPERTIES OF 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 ωi(k) or kj(ω) of the dispersion relation W(ω, k)=0. It is shown that solutions of initial value problems cannot consist of single wave modes if the initial values belong to W21(-∞, ∞); 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 kj(ω) lie in the upper half of the ω plane. The implications of this result for the Kramers-Kronig relations are discussed. © 1969 Martinus Nijhoff.