Question

Verify by substitution that U(x,t) = A e^B(x-vt) is a solution to the wave equation when A and B are constants.

Answer 1

The wave equation in one space dimension can be written like this:

Now,

U(x,t) = A e^B(x-vt)

dU/dt = Ae^B(x-vt)*(-Bv)

d²U/dt² = Ae^B(x-vt)*(-Bv)² = AB²v²e^B(x-vt)   .............(1)

Now, dU/dx = A e^B(x-vt)*B

d²U/dx² = Ae^B(x-vt)*(B)² = AB²e^B(x-vt)   .............(2)

From (1) and (2), we get

d²U/dt² = v²*(d²U/dx²)

Since, given equation U(x,t) = A e^B(x-vt) satisfies the wave equation criterion in one space dimension.

So,we can say that U(x,t) = A e^B(x-vt) is a solution to the wave equation when A and B are constants.