how do we describe this? 10. wave motion. an initial shape so can we just let the normal modes...

How do we describe this?

10. Wave Motion

An initial shape…

…so can we just let the normal modes oscillate at n?

No! Because that initial shape can do many things:

0 L

To get the answer right, we have to get all the boundary conditions right!

2

2

22

2 1ty

vxy

Wave equation needs two spatial and two

temporal boundary conditions.

Clamped string: y(0,t) = 0 “a certain point in space for all time” y(L,t) = 0

Two spatial boundary conditions

Two temporal boundary conditions

Initial shape: y(x,0) = “a certain point in time for all space”

rather than an analogous second point in time (which could work), try this:

Initial velocity: 0,xy

1

cossin,n

nnnn

n txv

Atxy

0,0 ty

0, tLy Lvn

n

0n

shape0, xy sAn '

The velocities tell you how the normal modes move

with different phase offset. But don’t do it this way!!!

0,xy

1

sincossin,n

nnnnn tAtBxv

txy

1

sin0,n

nn Bxv

xy

1

sinn

nBxLn French’s Fourier series!

(spatial boundary conditions)

1

cossinsin,n

nnnnnnn tAtBxv

txy

0sin0,1

nnn

n Axv

xy

Initially at rest?Then An = 0

0 10 20 30 40 50 60 70 80 90 100-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1