8.3: Wave Equations

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

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.