Answer
The wave equation is satisfied.
Work Step by Step
We need to compute the wave equation.
The wave equation can be defined as: $c^2 \dfrac{\partial^2 w}{\partial x^2}= \dfrac{\partial^2 w}{\partial t^2}$
In order to find the partial derivative, we will differentiate $w$ with respect to $t$, by keeping $x$ and $c$ as a constant, and vice versa:
$\dfrac{\partial w}{\partial t}=\dfrac{c}{x+ct}$ and $\dfrac{\partial w}{\partial x}= \dfrac{1}{x+ct}$
Now, $ \dfrac{\partial^2 w}{\partial t^2}= \dfrac{\partial}{\partial t}( \dfrac{1}{x+ct})= \dfrac{0-(c) (c)}{(x+ct)^2}=\dfrac{-c^2}{(x+ct)^2}$ ...(1)
and $c^2 \dfrac{\partial^2 w}{\partial x^2}=c^2 \dfrac{\partial}{\partial x}(\dfrac{1}{x+ct})=\dfrac{-c^2}{(x+ct)^2}$ ...(2)
Thus, equations (1) and (2) are equal and so the wave equation is satisfied.
