As before we need to consider three cases case 1 0 in

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Step II. We find the superposition of these solution which satisfies the nonhomogeneous initial conditions by means of Fourier analysis. To carry out Step I, we substitute u(x, t) = f (x)g (t) into the wave equation (4.18) and obtain f (x)g (t) = c2 f (x)g (t). We separate variables, putting all the functions involving t on the left, all the functions involving x on the right: g (t) f (x) = c2 . g (t) f (x) Once again, the left-hand side of this equation does not depend on x, while the right-hand side does not depend on t, so neither side can depend upon either x or t. Therefore the two sides must equal a constant λ, and our equation becomes g (t) f (x) = = λ, 2 g (t) c f (x) which separates into two ordinary differential equations, g (t) = λ, c2 g (t) or g (t) = λc2 g (t), (4.19) f (x) = λ, f (x) or f (x) = λf (x). (4.20) and Just as in the case of the heat equation, the homogeneous boundary condition u(0, t) = u(L, t) = 0 becomes f (0)g (t) = f (L)g (t) = 0, and assuming that g (t) is not identically zero, we obtain f (0) = f (L) = 0. Thu...
View Full Document

Ask a homework question - tutors are online