A Survey on Huygens’ Principle

Abstract

There are two classes of physical wave phenomena: with (e.g. water waves) or without (e.g. flash or bang in space) after effects. The second class is said to obey Huygens’ principle. The mathematical formulation concerns Cauchy’s initial value problem to a given linear hyperbolic differential equation, and is generalized to arbitrary dimensions as well as to curved spacetimes, i.e. Lorentzian manifolds. The original conjecture that every Huygens-type equation is transformable to the wave equation in Minkowski spacetime was refuted by counter-examples found by K. L. Steilmacher and by P. Günther. Since then, many results accumulated, but a general characterization of the equations which satisfy Huygens’ principle is not yet known. Some classes of examples show interesting relations to other branches of physics or mathematics: the new higher spinor field equations of Buchdahl and Wünsch solve the long-standing inconsistency problem, Huygensian wave equations on symmetric spaces are treated by means of Lie-theoretical methods, far-reaching generalizations of the Stellmacher-Lagnese examples are related to Coxeter groups and to integrable dynamical systems. The present paper surveys the research on Huygens’ principle - from Hadamard up to recent results.