  A number of equations is deduced which describe propagation of excitations
along $n$-dimensional surfaces in $R^N$. Usual excitations in wave theory
propagate along 1-dimensional trajectories. The role of the medium of
propagation of excitations considered in this paper is played by the infinite
dimensional space of $(n-1)$-dimensional surfaces in $R^N$. The role of rays is
played by $n$-dimensional solution surfaces of the variational problem. Such a
generalization of wave theory can be useful in quantum field theory. Among
these equations are the generalized Hamilton--Jacobi equation (known in
particular cases in the literature), generalized canonical Hamilton equations,
and generalized Schrodinger equation. Besides that, a theory of integration of
the generalized Hamilton--Jacobi equation is developed.
