Show that the function is a solution of the wave equation.
-w(x, t) = ln cxt

Question

Quizplus · Accepted Answer

The wave equation in two dimensions is given by $\frac{\partial^2 w}{\partial t^2} = c^2 \frac{\partial^2 w}{\partial x^2}$, where $c$ is a constant. For the function $w(x, t) = \ln(cxt)$, the second partial derivatives with respect to $x$ and $t$ do not satisfy this relationship. Specifically, $\frac{\partial^2 w}{\partial x^2} = -\frac{1}{x^2}$ and $\frac{\partial^2 w}{\partial t^2} = -\frac{1}{t^2}$, which do not satisfy the wave equation as their ratio is not a constant squared ($-\frac{1}{x^2} \neq c^2(-\frac{1}{t^2})$ for any constant $c$).

Show That the Function Is a Solution of the Wave