National Center for Physical Acoustics
The University of Mississippi1986
Light Diffraction and its Use in the Study of Acoustic Parametric
Oscillations
Light Diffraction and its Use in the Study of Acoustic Parametric
OscillationsMichael S. McPherson
University of MississippiNational Center for Physical
AcousticsUniversity, MS 38677
Michael S. McPhersonUniversity of Mississippi
National Center for Physical Acoustics
University, MS 38677
National Center for Physical Acoustics
The University of Mississippi1986
ColleaguesColleagues
• Mack Breazeale, Distinguished Professor of Research, UM, University, MS
• Alem Teklu, Ph. D., LSU, Asst. Prof., Charleston
• Mack Breazeale, Distinguished Professor of Research, UM, University, MS
• Alem Teklu, Ph. D., LSU, Asst. Prof., Charleston
National Center for Physical Acoustics
The University of Mississippi1986
Block Diagram of ApparatusBlock Diagram of Apparatus
National Center for Physical Acoustics
The University of Mississippi1986
National Center for Physical Acoustics
The University of Mississippi1986
Simultaneous Diffraction Patterns and Schlieren Images (a) without, (b) with Parametric Oscillation
Diffraction showing existence of subharmonicsDiffraction showing existence of subharmonics
National Center for Physical Acoustics
The University of Mississippi1986
Spectrum of Acoustical Parametric Oscillator (l0 = 7.5 cm)
National Center for Physical Acoustics
The University of Mississippi1986
Diffraction before and after the onset of of parametric
resonance at different frequencies and amplitudes
Diffraction before and after the onset of of parametric
resonance at different frequencies and amplitudes
National Center for Physical Acoustics
The University of Mississippi1986
Reflector Driver
l0
The Periodically Varying Cavity
National Center for Physical Acoustics
The University of Mississippi1986
Adler’s Data: Threshold Amplitude vs. Frequency
National Center for Physical Acoustics
The University of Mississippi1986
Current Data: Threshold Amplitude vs. frequency
National Center for Physical Acoustics
The University of Mississippi1986
l(t) =lo 1+hcos2ωt( )
Dissipative Wave Equation
∂2y∂t2
=c2∂2y∂x2 +
c3αω2
⎛ ⎝ ⎜ ⎞
⎠ ∂ 3y∂x2∂t
We assume
y =g(t)sinnπxl(t)
⎛ ⎝ ⎜ ⎞
⎠
National Center for Physical Acoustics
The University of Mississippi1986
Time−dependent Part (after expansion):
d2gdz2
+aαcω
⎛ ⎝
⎞ ⎠ dgdz
⎛ ⎝
⎞ ⎠
+ a−2qcos2z( )g=0
Where
z =ωt, a=ωn
2
ω2 , q=ah
Mathieu's Equation
National Center for Physical Acoustics
The University of Mississippi1986
Transformed into Mathieu's equation by substitution
G(z)=g(z)exp −aαcω
⎛
⎝⎜⎞
⎠⎟z
⎡
⎣⎢
⎤
⎦⎥
then,
d2G dz2 + a−2qcos2z( )G =0
where
a= a-a2α 2c2
ω 2
⎛
⎝⎜⎞
⎠⎟
Solution, a damped Mathieu function, can be stable, unstable, or neutral, depending on values of a and q.
National Center for Physical Acoustics
The University of Mississippi1986
National Center for Physical Acoustics
The University of Mississippi1986
National Center for Physical Acoustics
The University of Mississippi1986
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Regions of instability :
hω ≥ ωn −ω( )2
+ aαc( )2
a =1, 4, 9, 16, 25, etc...
National Center for Physical Acoustics
The University of Mississippi1986
Conclusions
• Results of a new investigation of parametric resonance in an acoustic resonator are presented
• Measurements of the threshold drive amplitude frequency dependence show an apparent contradiction with previous results
• Resonators with differing lengths can result in the onset of parametric amplification in different regions of instability
• These differing regions of instability account for the differences in the observed data trends
