Some properties of the solutions of wave equations

✍ Scribed by Broer, L. J. F. ;Peletier, L. A.

• Publisher: Springer
• Year: 1969
• Tongue: English
• Weight: 991 KB
• Volume: 21
• Category: Article
• ISSN: 0003-6994
• DOI: 10.1007/bf00411602

✦ Synopsis

Some properties of solutions of initial value problems and mixed initialboundary 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 coi(k) or kj(~o) of the dispersion relation W(co, k) ~ O. It is shown that solutions of initial value problems cannot consist of single wave modes if the initial values belong to WI(--o% c~); generally such solutions must contain all possible modes. Similar results hold for solutions of mixed initial-boundary value problems. I t is found that such solutions are stable, even if some of the singularities of the functions ki(~o) lie in the upper half of the ~o plane. The implications of this result for the Kramers-Kronig relations are discussed. § 1. I n t r o d u c t i o n