8.3: Wave Equations
( \newcommand{\kernel}{\mathrm{null}\,}\)
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[∂u∂t]=∂∂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[∂u∂x]=∂∂x[15πcos(3πx)sin(4πy)cos(10πt)]=−45π2sin(3πx)sin(4πy)cos(10πt)
uyy(x,y,t)=∂∂y[∂u∂y]=∂∂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.