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)

**d ^{2}U/dt^{2} =
Ae^{B(x-vt)}*(-Bv)^{2} =
AB^{2}v^{2}e^{B(x-vt)
}.............(1)**

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

**d ^{2}U/dx^{2} =
Ae^{B(x-vt)}*(B)^{2} = AB^{2}e^{B(x-vt)
}.............(2)**

From (1) and (2), we get

**d ^{2}U/dt^{2} =
v^{2}*(d^{2}U/dx^{2})**

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.

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