The evolution of short gravity waves on long gravity waves on the surface of deep water is studied. Both wave trains are assumed to be irrotational, mild in slope, and slowly modulated in space and time, but their scales are so different that the short wavelength is very much less than the long-wave amplitude. Here, it is shown that the use of Lagrangian instead of the usual Eulerian coordinates is advantageous for yielding analytical results. Linear and nonlinear evolutions of short waves over intermediate and very long distances are discussed.