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:
$\dfrac{\partial w}{\partial t}=c \cos (x+ct)-2 c\sin (2x+2ct) \\ \dfrac{\partial w}{\partial x}= \cos (x+ct)-2 \sin (2x+2ct)$
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)$
and $c^2 \dfrac{\partial^2 w}{\partial x^2}=-c^2 \sin (x+ct)-4c^2 \cos (2x+2ct)$
So, $c^2 \dfrac{\partial^2 w}{\partial x^2}= \dfrac{\partial^2 w}{\partial t^2}$
Thus, it has been proved that the wave equation is satisfied.
