Answer
The wave equation is satisfied.
Work Step by Step
We need to prove the wave equation $c^2 \dfrac{\partial^2 w}{\partial x^2}= \dfrac{\partial^2 w}{\partial t^2}$
We take the first partial derivative of the given function $w(t,x)$ with respect to $x$, by treating $t$ as a constant, and vice versa:
$\dfrac{\partial w}{\partial t}=\dfrac{c}{x+ct} \\ \dfrac{\partial w}{\partial x}= \dfrac{1}{x+ct}$
Now, $ \dfrac{\partial^2 w}{\partial t^2}= \dfrac{0-(c) (c)}{(x+ct)^2}=\dfrac{-c^2}{(x+ct)^2}$
and $c^2 \dfrac{\partial^2 w}{\partial x^2}=c^2 \times \dfrac{\partial}{\partial x}(\dfrac{1}{x+ct})=-\dfrac{c^2}{(x+ct)^2}$
Thus, it has been proved that the wave equation is satisfied.
