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:

(23)#\[ u(x,0)=A e^{i k x}, \]

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

(24)#\[ u(x,t)=G(t)H(x). \]

Substituting in (22), we have

\[\begin{align*} \frac{\partial }{\partial t} (G(t)H(x))=& -c \frac{\partial }{\partial x} (G(t)H(x)) \\ H(x) \frac{\partial G(t)}{\partial t} =& -c G(t) \frac{\partial H(x)}{\partial x} \\ \frac{1}{G(t)} \frac{\partial G(t)}{\partial t} =& -c \frac{1}{H(x)} \frac{\partial H(x)}{\partial x}, \end{align*}\]

that to be valid for all \((x,t)\) must be equal to a constant \(-\alpha\). Integration yields

\[\begin{align*} \frac{1}{G(t)} \frac{\partial G(t)}{\partial t} =& -\alpha \Leftrightarrow G(t)=A_1e^{-\alpha t} \\ -c \frac{1}{H(x)} \frac{\partial H(x)}{\partial x}=& -\alpha \Leftrightarrow H(x)=A_2e^{\alpha x/c}, \end{align*}\]

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:

(25)#\[ u(x,t)=Ae^{-ikct}e^{ikcx/c}=Ae^{ik(x-ct)}, \]

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 \(c\).