Page HW Class Topic

17 2.19cdg 2.19e Classification

17 2.20 Classification

17 2.21bc 2.21a Classification: take p=p(x,y,z)!

18 2.24 2.26 Canonical form

18 2.25 Canonical form

18 2.22cgh 2.22d Characteristics

18 2.27ab 2.27d 2D Canonical form

18 2.28ajkp 2.28nml 2D Canonical form

98 7.19 7.20 Unsteady heat conduction in a bar

98 7.21 7.22 Unsteady heat conduction in a bar

98 7.24 7.25 Unidirectional viscous flow

98 7.26 Unidirectional viscous flow

99 7.28 7.27 Acoustics in a pipe

99 7.31 7.29 Vibrations of a string

99 7.35 7.36 Acoustics above a plate

99 7.37 7.37 Steady heat conduction in a plate

99 7.38 7.39 Steady heat conduction in a disk 4|c| Solve the problem below

1.

Solve the unsteady vibrations of the membrane on a circular drum. The membrane displacement $u(r,\theta,t)$ satisfies the wave equation

$$u_{tt} = a^2 \nabla^2 u.$$

The boundary condition at the edge is

$$u(2,\theta,t)= 0.$$

For initial condition, assume that the membrane is struck at the point r=1 and $\theta=\pi/2$. In other words assume that $u(r,\theta,0)=0$ and that $u_t(r,\theta,0)$ is a delta function at r=1 and $\theta=\pi/2$. This means that

$$\int_0^1 \int_0^{2\pi} u_t(r,\theta,0) f(r,\theta)\/ r \/{\rm d} r\/ {\rm d}\theta = f(1,\pi/2)$$

for any function $f(r,\theta)$. What frequencies will be present in the vibrations, and with what relative strength?