T T Standing Waves Progressive Waves Stretched string Newton’s 2nd Wave Equation.
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Transcript of T T Standing Waves Progressive Waves Stretched string Newton’s 2nd Wave Equation.
T
T
2
2
22
2 1ty
vxy
maF
Standing Waves Progressive Waves
tL
xnn
cossin
vtxf
Stretched string
Newton’s 2ndWave Equation
Any wave motion on the string can be described by a sum of these modes!
Fourier Analysis
Normal Modes:
n = 1 n = 2 n = 3
Joseph Fourier, 1768-1830
One equation, infinity unknowns …
…thanks for nothin’ Joe!
1
sinn
n Lxn
Bxy
Focus on the shape first! y(x)
How to find a specific Bn?
L
nn
L
dxL
xnL
xnBdx
Lxn
xy0 1
*
0
*
sinsinsin
coscossinsin 21
dx
Lxnn
LxnnB
n
Ln
1 0
**
coscos2
This reduces series to 1 term!
Mathematically…
Multiply by nth harmonic and integrate.
L
n
n
Lxnn
nnL
Lxnn
nnLB
01
*
*
*
* sinsin2
1
**
** 00sinsin
2n
n nnnnL
nnnnLB
*nn
*nn
0 0 …all terms zero!
…uh oh….0 00
xgxf
xgxf
cxcx
limlim
L’Hopital’s Rule:
If f(c)=0 and g(c)=0 then
x
10cos
2
* LB
n
2
*LBn
2
sin*
0
* LBdx
Lxn
xy nL
dxL
xnxy
LB
L
n
0
sin2
1
sinn
n Lxn
Bxy
Any shape y(x) between 0 and L
n=1 n=2 n=3 n=4 n=5 n=6
Positive contribution
zero contribution
zero contribution
zero contribution
zero contribution
zero contribution
zero contribution
Positive contribution
zero contribution
zero contribution
zero contribution
zero contribution
Graphically….
0sinsin0
21
L
dxL
xnL
xn “orthogonal”
n*=1
n*=2
0 L
dxL
xncx
LB
L
n
0
sin2
Integrate by parts:
b
a
ba
b
a
dxxgxfxgxfdxxgxf
LL
n Lxn
nL
Lxn
nL
xLc
B00
coscos2
c = slope
LL
n Lxn
nL
Lxn
nL
xLc
B00
coscos2
L
n Lxn
nL
Lxn
xn
cB
0
sincos2
000cos2
nLn
cBn
nncL
Bn cos2
1
sinn
n Lxn
Bxy
clearload f;f=f-172;plot(f)B(200)=0;y(355)=0;for n = 1:200 for x = 1:355 B(n)=B(n)+(2/355)*f(x)*sin(n*pi*x/355); endend
figureplot(B,'.')for x = 1:355 for n = 1:200 y(x)=y(x)+B(n)*sin(n*pi*x/355); endendfigureplot(y)
(French’s Fourier Foibles)
1
sinn
n Lxn
Bxy dx
Lxn
xyL
BL
n
0
sin2
Good choice if boundaries are at 0: Not so good for other shapes..
Les Foibles de Fourier en Frances
More General: function periodic on interval x = –L to x = L.
11
0 sincos2 n
nn
n Lxn
BL
xnA
Axy
dxL
xnxy
LB
L
L
n
sin1
dxL
xnxy
LA
L
L
n
cos1
dxxyL
AL
L
10
Something hard:
0 L/8 L
H
x
L
L
L
dxL
HdxL
A8
8
0
0 011
80
HA
L
L
L
n dxL
xnL
dxL
xnH
LA
8
8
0
cos01
cos1
8
0
sinL
n Lxn
nL
LH
A
8
sin
n
nH
An
0sin1
8
0
L
n dxL
xnH
LB
8
0
cosL
n Lxn
nL
LH
B
8
cos1
n
nH
Bn
11
sin8
cos1cos8
sin8 nn L
xnnnH
Lxnn
nHH
xy