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8.3: Wave Equations

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utt=c2(uxx+uyy)wave equation in two dimensions

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 8.3.1: A Solution to the Wave Equation

Verify that

u(x,y,t)=5sin(3πx)sin(4πy)cos(10πt)

is a solution to the wave equation

utt=4(uxx+uyy).

Solution

First, we calculate utt,uxx, and uyy:

utt(x,y,t)=t[ut]=t[5sin(3πx)sin(4πy)(10πsin(10πt))]=t[50πsin(3πx)sin(4πy)sin(10πt)]=500π2sin(3πx)sin(4πy)cos(10πt)

uxx(x,y,t)=x[ux]=x[15πcos(3πx)sin(4πy)cos(10πt)]=45π2sin(3πx)sin(4πy)cos(10πt)

uyy(x,y,t)=y[uy]=y[5sin(3πx)(4πcos(4πy))cos(10πt)]=y[20πsin(3πx)cos(4πy)cos(10πt)]=80π2sin(3πx)sin(4πy)cos(10πt).

Next, we substitute each of these into the right-hand side of Equation ??? and simplify:

4(uxx+uyy)=4(45π2sin(3πx)sin(4πy)cos(10πt)+80π2sin(3πx)sin(4πy)cos(10πt))=4(125π2sin(3πx)sin(4πy)cos(10πt))=500π2sin(3πx)sin(4πy)cos(10πt)=utt.

This verifies the solution.


This page titled 8.3: Wave Equations is shared under a not declared license and was authored, remixed, and/or curated by Pamini Thangarajah.

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