The classical method of characteristics is a powerful tool for construction of smooth solutions to nonlinear first order PDEs. Certain generalization of this approach (method of singular characteristics (MSC)) is useful for the construction of the surfaces where the solution is non-smooth. In this paper it is shown that the MSC can be used for the construction of singular surfaces (weak waves) in some second order PDEs – Euler-Lagrange equation for multiple integral variational problem. A two dimensional variational wave equation is considered as an example. The phenomenon of bifurcation of the weak waves (singular lines)is found using analytical and numerical methods.