Standing Waves

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Stretched string
Wave Equation
Newtons 2nd
Progressive Waves
Standing Waves
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Fourier Analysis
Normal Modes
n 1
n 2
n 3
Any wave motion on the string can be described by
a sum of these modes!
One equation, infinity unknowns thanks for
nothin Joe!
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Focus on x first!
How to find a specific Bn?
Multiply by nth harmonic and integrate.
This reduces series to 1 term!
Mathematically
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all terms zero!
uh oh.
0
x
LHopitals Rule
If f(c)0 and g(c)0 then
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Any shape between 0 and L
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Graphically.
n6
n5
n4
n3
n2
n1
n1
Positive contribution
zero contribution
zero contribution
zero contribution
zero contribution
zero contribution
n2
zero contribution
Positive contribution
zero contribution
zero contribution
zero contribution
zero contribution
orthogonal
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0
L
c slope
Integrate by parts
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1st
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1st 2nd
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1st 2nd 3rd
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1st 2nd 3rd 4th
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1st 2nd 3rd 4th 5th
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Barrys profile
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Barrys profile
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Barrys 1st harmonic
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Barrys 1st and 2nd harmonic
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Barrys 1st, 2nd, and 3rd harmonic
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clear load f ff-172 plot(f) B(200)0 y(355)0
for n 1200 for x 1355
B(n)B(n)(2/355)f(x)sin(npix/355)
end end figure plot(B,'.') for x 1355 for
n 1200 y(x)y(x)B(n)sin(npix/355)
end end figure plot(y)
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Frenchs Fourier Foible
Good choice if boundaries are at 0
Not so good for other shapes..
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More General function periodic on interval x
L to x L.
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Something hard
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MATLAB summation of 1 to 100 terms.
H
0
0
L
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Time dependence?
Make y(x) an initial condition y(x,0) y(x)
The normal modes will oscillate at wn
Given the wavelengths, the frequencies are known.
This is NOT a consequence of Fourier Analysis
This IS a consequence of the wave equation
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18,000,000 sinusoids in MATLAB
H
0
0
L
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Any well behaved repetitive function can be
described as an infinite sum of sinusoids with
variable amplitudes (a Fourier Series). On a
stretched string these correspond to the normal
modes. Fourier analysis can describe arbitrary
string shapes as well as progressive waves and
pulses.