PHY 711 Fall 2013 -- Lecture 29 111/11/2013
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 29 --
Chap. 9: Wave equation for sound
1. Standing waves
2. Green’s function for wave equation; wave scattering
PHY 711 Fall 2013 -- Lecture 29 211/11/2013
PHY 711 Fall 2013 -- Lecture 29 311/11/2013
PHY 711 Fall 2013 -- Lecture 29 411/11/2013
PHY 711 Fall 2013 -- Lecture 29 511/11/2013
Comments about exam
Note that this figure is misleading; a>b for physical case
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22
2
202
2
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sin2
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sin
PHY 711 Fall 2013 -- Lecture 29 611/11/2013
Comment about exam – continued
mgkT
pss
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dp
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PHY 711 Fall 2013 -- Lecture 29 711/11/2013
Comment about exam – continued
PHY 711 Fall 2013 -- Lecture 29 811/11/2013
Linearization of the fluid dynamics relations:
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v
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PHY 711 Fall 2013 -- Lecture 29 911/11/2013
0
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PHY 711 Fall 2013 -- Lecture 29 1011/11/2013
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PHY 711 Fall 2013 -- Lecture 29 1111/11/2013
Time harmonic standing waves in a pipe
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PHY 711 Fall 2013 -- Lecture 29 1211/11/2013
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PHY 711 Fall 2013 -- Lecture 29 1311/11/2013
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PHY 711 Fall 2013 -- Lecture 29 1411/11/2013
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PHY 711 Fall 2013 -- Lecture 29 1511/11/2013
)( :functions Bessel xJm
m=0
m=1
m=2
PHY 711 Fall 2013 -- Lecture 29 1611/11/2013
m=0
m=1
m=2
)(
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xdJm
Zeros of derivatives: m=0: 0.00000, 3.83171, 7.01559 m=1: 1.84118, 5.33144, 8.53632 m=2: 3.05424, 6.70613, 9.96947
PHY 711 Fall 2013 -- Lecture 29 1711/11/2013
Boundary condition for z=0, z=L:
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sin)( 0)()0(
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PHY 711 Fall 2013 -- Lecture 29 1811/11/2013
Example
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PHY 711 Fall 2013 -- Lecture 29 1911/11/2013
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PHY 711 Fall 2013 -- Lecture 29 2011/11/2013
01
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PHY 711 Fall 2013 -- Lecture 29 2111/11/2013
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PHY 711 Fall 2013 -- Lecture 29 2211/11/2013
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PHY 711 Fall 2013 -- Lecture 29 2311/11/2013
Derivation of Green’s function for wave equation
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PHY 711 Fall 2013 -- Lecture 29 2411/11/2013
Derivation of Green’s function for wave equation -- continued
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PHY 711 Fall 2013 -- Lecture 29 2511/11/2013
Derivation of Green’s function for wave equation -- continued
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PHY 711 Fall 2013 -- Lecture 29 2611/11/2013
Derivation of Green’s function for wave equation -- continued
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PHY 711 Fall 2013 -- Lecture 29 2711/11/2013
Derivation of Green’s function for wave equation – continued need to find A and B.
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PHY 711 Fall 2013 -- Lecture 29 2811/11/2013
Derivation of Green’s function for wave equation – continued
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PHY 711 Fall 2013 -- Lecture 29 2911/11/2013
Derivation of Green’s function for wave equation – continued
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PHY 711 Fall 2013 -- Lecture 29 3011/11/2013
For time harmonic forcing term we can use the corresponding Green’s function:
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In fact, this Green’s function is appropriate for boundary conditions at infinity. For surface boundary conditions where we know the boundary values or their gradients, the Green’s function must be modified.
PHY 711 Fall 2013 -- Lecture 29 3111/11/2013
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PHY 711 Fall 2013 -- Lecture 29 3211/11/2013
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PHY 711 Fall 2013 -- Lecture 29 3311/11/2013
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PHY 711 Fall 2013 -- Lecture 29 3411/11/2013
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PHY 711 Fall 2013 -- Lecture 29 3511/11/2013
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PHY 711 Fall 2013 -- Lecture 29 3611/11/2013
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PHY 711 Fall 2013 -- Lecture 29 3711/11/2013
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PHY 711 Fall 2013 -- Lecture 29 3811/11/2013
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