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:
$\dfrac{\partial w}{\partial t}=c \cos (x+ct)-2 c\sin (2x+2ct)$ and $\dfrac{\partial w}{\partial x}= \cos (x+ct)-2 \sin (2x+2ct)$
Now, $ \dfrac{\partial^2 w}{\partial t^2}= \dfrac{\partial}{\partial t}(c \cos (x+ct)-2 c\sin (2x+2ct))=-c^2 \sin (x+ct)-4c^2 \cos (2x+2ct)$ ...(1)
and $c^2 \dfrac{\partial^2 w}{\partial x^2}=c^2 \dfrac{\partial}{\partial x}(\cos (x+ct)-2 \sin (2x+2ct))=-c^2 \sin (x+ct)-4c^2 \cos (2x+2ct)$ ...(2)
Thus, equations (1) and (2) are equal, so the wave equation is satisfied.
