This work is the outgrowth of a plan to make a uniform presentation of the investigations on earthquake seismology, underwater sound, and model seismology carried on by the group connected with Lamont Geological Observatory of Columbia University. The scope was subsequently enlarged to cover a particular selection of related problems. The methods and results of the theory of wave propagation in layered media are important in seismology, in geophysical prospecting, and in many problems of acoustics and electromagnetism.