Upload
lionel-bruce
View
212
Download
0
Embed Size (px)
DESCRIPTION
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:
Citation preview
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