light diffraction and its use in the study of acoustic parametric oscillations

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

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The University of Mississippi1986

Adler’s Data: Threshold Amplitude vs. Frequency

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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

€

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