PP-wave strings from membrane and from string in the spacetime with two time directions

In this Letter we obtain strings that propagate in the quantized pp-wave backgrounds. We can obtain these strings from the solutions of membrane. The other way is the propagation of a massless string in a spacetime with two time dimensions. This string sweeps a worldvolume, which enables us to obtain other strings in the quantized pp-wave backgrounds in the spacetime with one time direction. The associated algebras and Hamiltonians of these massive strings will be studied.