Answer
The wave equation is satisfied.
Work Step by Step
We need to compute the wave equation.
The wave equation can be defined as: $c^2 \dfrac{\partial^2 w}{\partial x^2}=-c^2 \sin (x+ct)$
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:
$\dfrac{\partial w}{\partial t}=c \cos (x+ct)$ and $\dfrac{\partial w}{\partial x}=\cos (x+ct)$
Now, $c^2 \dfrac{\partial^2 w}{\partial x^2}=c^2 \dfrac{\partial}{\partial x}(\dfrac{\partial w}{\partial x})=c^2 [-\sin (x+ct)]=-c^2 \sin (x+ct)$
Thus, the wave equation is satisfied.
