8.3: Wave Equations

$$\underset{\text{wave equation in two dimensions}}{u_{tt}=c^2(u_{xx}+u_{yy})}$$

The unknown function $u$ has three independent variables: $t$, $x$, and $y$ with $c$ is an arbitrary constant. The independent variables $x$ and $y$ are considered to be spatial variables, and the variable $t$ represents time.

Example $\PageIndex{1}$: A Solution to the Wave Equation

Verify that

$$u(x,y,t)=5\sin(3πx)\sin(4πy)\cos(10πt)$$

is a solution to the wave equation

$$u_{tt}=4(u_{xx}+u_{yy}). \label{Ex7Eq2}$$

Solution

First, we calculate $u_{tt},u_{xx},$ and $u_{yy}:$

$$\begin{align} u_{tt}(x,y,t) &=\dfrac{∂}{∂t}\left[\dfrac{∂u}{∂t}\right] \nonumber \\[6pt] &=\dfrac{∂}{∂t}[5\sin(3πx)\sin(4πy)(−10π\sin(10πt))] \nonumber \\[6pt] &= \dfrac{∂}{∂t} \left[−50π\sin(3πx)\sin(4πy)\sin(10πt)\right] \nonumber \\[6pt] &=−500π^2\sin(3πx)\sin(4πy)\cos(10πt) \nonumber \\ & \end{align}$$

$$\begin{align} u_{xx}(x,y,t) &=\dfrac{∂}{∂x} \left[\dfrac{∂u}{∂x}\right] \nonumber \\[6pt] &=\dfrac{∂}{∂x}\left[15π\cos(3πx)\sin(4πy)\cos(10πt)\right] \nonumber \\[6pt] &=−45π^2\sin(3πx)\sin(4πy)\cos(10πt) \end{align}$$

$$\begin{align} u_{yy}(x,y,t) &=\dfrac{∂}{∂y} \left[\dfrac{∂u}{∂y} \right] \nonumber \\[6pt] & =\dfrac{∂}{∂y}\left[5\sin(3πx)(4π\cos(4πy))\cos(10πt)\right] \nonumber \\[6pt] &=\dfrac{∂}{∂y}\left[20π\sin(3πx)\cos(4πy)\cos(10πt)\right] \nonumber \\ &=−80π^2\sin(3πx)\sin(4πy)\cos(10πt). \end{align}$$

Next, we substitute each of these into the right-hand side of Equation $\ref{Ex7Eq2}$ and simplify:

$$\begin{align} 4(u_{xx}+u_{yy})&=4(−45π^2\sin(3πx)\sin(4πy)\cos(10πt)+−80π^2\sin(3πx)\sin(4πy)\cos(10πt)) \nonumber \\[6pt] &=4(−125π^2\sin(3πx)\sin(4πy)\cos(10πt)) \nonumber \\[6pt] &=−500π^2\sin(3πx)\sin(4πy)\cos(10πt) \nonumber \\[6pt] & =u_{tt}. \end{align}$$

This verifies the solution.