how do we describe this? 10. wave motion. an initial shape so can we just let the normal modes...
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:TRANSCRIPT
![Page 1: 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](https://reader036.vdocument.in/reader036/viewer/2022090102/5a4d1bac7f8b9ab0599cb37b/html5/thumbnails/1.jpg)
How do we describe this?
10. Wave Motion
![Page 2: 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](https://reader036.vdocument.in/reader036/viewer/2022090102/5a4d1bac7f8b9ab0599cb37b/html5/thumbnails/2.jpg)
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!
![Page 3: 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](https://reader036.vdocument.in/reader036/viewer/2022090102/5a4d1bac7f8b9ab0599cb37b/html5/thumbnails/3.jpg)
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
![Page 4: 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](https://reader036.vdocument.in/reader036/viewer/2022090102/5a4d1bac7f8b9ab0599cb37b/html5/thumbnails/4.jpg)
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
![Page 5: 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](https://reader036.vdocument.in/reader036/viewer/2022090102/5a4d1bac7f8b9ab0599cb37b/html5/thumbnails/5.jpg)
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
![Page 6: 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](https://reader036.vdocument.in/reader036/viewer/2022090102/5a4d1bac7f8b9ab0599cb37b/html5/thumbnails/6.jpg)
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