The exact equation
The exact equation#
The analytic (exact) solution of (22) can be obtained by the method of the separation of variables. We shall assume an initial condition in the form of a simple wave:
where \(k = 2\pi/\lambda\) is the wave number (\(\lambda\) is the wavelength).
Let the solution to (22) with initial condition (23) be given by
Substituting in (22), we have
that to be valid for all \((x,t)\) must be equal to a constant \(-\alpha\). Integration yields
and we find \(u(x,t)=G(t)H(x)=A_1A_2e^{-\alpha t}e^{\alpha x/c}\).
At \(t=0\) we have \(u(x,0)=A_1A_2e^{\alpha x/c}\), so we know that \(A=A_1A_2\) and \(ik = \alpha/c\) and so \(\alpha=ikc\). Finally, we can write:
which is the solution of (22) given the initial condition (23). This solution represents the initial condition moving along the positive \(x\)-direction with translation velocity \(c\).