Determine whether the function is a solution of the wave equation.
-w(x, t) = cos ( 4ct) sin ( 4x)

Question

Quizplus · Accepted Answer

The wave equation in two dimensions is given by $\frac{\partial^2 w}{\partial t^2} = c^2 \frac{\partial^2 w}{\partial x^2}$. For $w(x, t) = \cos(4ct) \sin(4x)$, taking the second partial derivative with respect to $t$ gives $-16c^2 \cos(4ct) \sin(4x)$, and taking the second partial derivative with respect to $x$ gives $-16 \cos(4ct) \sin(4x)$. Multiplying the second spatial derivative by $c^2$ gives $-16c^2 \cos(4ct) \sin(4x)$, which matches the second time derivative, thus satisfying the wave equation.

Determine Whether the Function Is a Solution of the Wave