Characterization of High Q Spherical Resonators

National Center for Physical Acoustics1986

The University of Mississippi 1 8 4 8

Characterization of High Q Spherical Resonators

Kenneth Bader, Jason Raymond, Joel MobleyNational Center for Physical Acoustics

University of Mississippi

Felipe Gaitan, Ross Tessien, Robert HillerImpulse Devices, Inc.

Grass Valley, CA

Physics Colloquium25 March, 2008



Outline• Introduction/Motivation

–acoustic cavitation–sonoluminescence

• Measurements in test resonator –determination acoustic modes

• Comparison of theory and measurements–shell vibration accommodation

• Measurements in High Q resonator



• Cavitation is excitation of bubbles using acoustics

• Bubble's radial symmetry effective at concentrating energy

• Sonoluminescence– violent bubble collapse– bubble wall supersonic– gas temperature ~ 105 K– energy concentration 12

orders magnitude– picosecond flashes of light

Bubble action under acoustic source

Single bubble sonoluminescence(30 kHz, Water)

Introduction

pa

pr

pc



Motivation

• Project goal is maximize cavitation collapse – how hot will the gas get?

• Methods for maximizing cavitation collapse– liquids which promote violent collapse– higher ambient pressure of system– high Q resonators



High Q Resonator for Acoustic Cavitation• Resonator Properties

– 10” OD stainless steel sphere– excited with single acoustic horn– fittings for high pressures– Q > 10 000

• Characterization of Resonator– determine acoustic modes

• laser dopper vibrometry (LDV)• surface piezoelectric transducer• piezoelectric transducer in fluid

(hydrophone)



Acoustic Modes• Determine velocity potential

from Helmholtz equation

• are spherical Bessel

– jo mode of interest

• Boundary condition at sphere edge

– for infinitely rigid boundary

∇2k 2 =0

i ur=a=−∂pr=a

∂ r

kaj n /

j o

jn for n = 0-3

ur=a= 0kn, s=zn,s /a

Pn∝ jnkn ,s r Y n,m

un∝ j 'n kn, s r Yn ,m



Cell Hydrophone

Automated Positioning System

Set-up for Cell Pressure Measurements



Normalized Pressure vs. Position, Frequency

Frequency (kHz)

Dis

tan

ce f

rom

SL

Cel

l Cen

ter

(mm

)

Hydrophone Response in Cell



Normalized Pressure vs. Position, Frequency

Frequency (kHz)

Dis

tan

ce f

rom

SL

Cel

l Cen

ter

(mm

)

Distance from SL Cell Center (mm)

Hydrophone Response in Cell



Hydrophone Response in CellNormalized Pressure vs. Position, Frequency

Frequency (kHz)

Dis

tan

ce f

rom

SL

Cel

l Cen

ter

(mm

)

Frequency (kHz)



• Motion of shell dictated by elasticity dynamics:

– s is shell displacement

– cl is longitudinal sound speed

– ct is transverse sound speed

• Shell motion modifies acoustic boundary condition as

−ω2s=cl2∇∇⋅s−ct

2∇ x∇ xs

j 'n ka =−c2

sc l2 ka jn ka Sn k la

Mehl, J. 1985, J. Acoust. Soc. Am. 78, 782.

ur=a= j 'n ka = 0

Comparison with Theory

Infinitively Rigid BC

Elastic Shell BC



Comparison with Theory • Motion of shell dictated by elasticity dynamics:

– s is shell displacement

– cl is longitudinal sound speed

– ct is transverse sound speed

• Shell motion modifies acoustic boundary condition as

• Solve for eigenfrequencies ka

−ω2s=cl2∇∇⋅s−ct

2∇ x∇ xs

j 'nka =−c2

sc l2 ka jn ka Snk la

Mehl, J. 1985, J. Acoust. Soc. Am. 78, 782.

Sn kla ∝particlevelocity

pressure



'

Calculated Acoustic Modesj ' 0 ka , −

c2

s c l2 ka j 0 ka S0 k l a vs. ka

ka



Calculated Acoustic Modes

Acoustic Mode 130.4 kHz

j ' 0 ka , −c2

s c l2 ka j 0 ka S0 k l a vs. ka


ka

'




Calculated Empty Shell Breathing Modes

kla

So

Shell Mode48.5 kHz

Divergence of So



Measured Amplitude of Acoustic, Shell Modes

Acoustic Mode 130.4kHz


Shell Mode49.0 kHz

Frequency (kHz)

Pressure Profile at Cell Center vs. FrequencyH

ydro

ph

one

Res

pons

e (m

V)



10” Resonator Measurements Experimental Setup

Experimental arrangement used to measure acoustic pressure within the resonator

SpectrumAnalyzer

PowerAmplifier

Resonator

HydrophoneAutomatedPositioningSystem

Tracking Gen.OUT

IN

SpectrumAnalyzer

PowerAmplifier

Resonator

HydrophoneAutomatedPositioningSystem

Tracking Gen.OUT

IN



Resonance Frequencies - Q of Resonance

Resonance Half-width (Frequency) 17.8 kHz (0,4) 38.0 Hz 24.2 kHz (0,5) 8.3 Hz 31.0 kHz (0,6) 4.9 Hz 37.9 kHz (0,7) 1.8 Hz

Summary of Measured Resonance Frequencies 25 2530

20

10

0

Frequency (Hz)

Ampl

itude

(dB)

Frequency Response Near Resonance

(0,4)

(0,5)

(0,6)

(0,7)



50 40 30 20 10 0 10 20 30 40 500

0.5

1

Distance (mm)

Rela

tive

Inte

nsity

Intensity vs. Radius

(0,4)(0,5)(0,6)(0,7)

Pressure vs. Radius

Relative amplitude of acoustic pressure for the radial acoustic (n=0) modes

Pressure Half-maximum (Distance) 17.8 kHz (0,4) 38 mm 24.2 kHz (0,5) 25 mm 31.0 kHz (0,6) 18 mm 37.9 kHz (0,7) 15 mm

24.1 cm

10 cm

24.1 cm

10 cm

50 25 0 25 500

1

Distance (mm)

Rela

tive

Inte

nsity

Intensity vs. Radius

(0,4)

MeasuredTheory



Summary

• Maximizing cavitation collapse involves the use of high Q resonators

• Preliminary work requires developing techniques for characterizing high Q resonators

• Initial studies done in sonoluminescence cell – zero order acoustic modes correlate well with Mehl theory– empty shell breathing modes correlate well with Mehl theory

• 10” Resonator resonant frequencies, pressure profiles agree with theory – Q range from ~ 400 to 20 000



Acknowledgements• The authors would like to thank members

of the Ultrasonics Group at NCPA for their help with this work.

• This work was supported by SMDC Contract NO. W9113M-07-C-0178



The End



Passive Cavitation Detector Response

24 24.05 24.1 24.15 24.20

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency (kHz)Re

spon

se (V

pp)

Cavitation Detector Response vs. Frequency

↑ (0,4)

Cavitation detector response measured near resonance Measurement based on high-

frequency emissions from collapsing bubbles

Passive acoustic sensor mounted near wall of resonator

Amplitude of high-pass filtered signal from PZT-pin transducer mounted near wall of resonator

FunctionGenerator

PZT-pin sensor

Oscilloscope

H.P. Filter(400 kHz)

PowerAmplifier

Resonator

FunctionGenerator

PZT-pin sensor

Oscilloscope

H.P. Filter(400 kHz)

PowerAmplifier

Resonator



Radial Mode Resonance Frequencies -Temperature Dependence

Acoustic resonance frequencies measured at different temperatures agree well with those predicted by the theory

~ 45 Hz/°C dependence (near room temperature)

0 5 10 15 20 25 30 35 40 45 5022

24

26

28

30

32

34

36

38

40

Temperature (deg C)

Freq

uenc

y (k

Hz)

Resonance Frequency vs. Temperature

(0,4)

(0,5)

(0,6)

Predicted (-) and measured (^) acoustic resonance frequencies vs. temperature

30.8 30.85 30.9 30.95 31 31.05 31.1 31.15 31.2 31.25 31.320

10

0

(0,6) 20oC(0,6) 22oC

(0,6) 25oC

Frequency (kHz)

Ampl

itude

(dB)

Frequency Response Variation With Temperature



Acoustic Resonances – Rigid Approximation

Radial Modes

(n=0)

0 5 10 150

10

20

30

40

50

60

70

80

Mode Number, n

Freq

uenc

y (k

Hz)

Frequency Spectrum Rigid Approximation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.2

0

0.2

0.4

0.6

0.8

1

Radial distance (r/a)

Acou

stic

Pre

ssur

e (p

/p0)

Radial Acoustic Modes

(0,2)

(0,3)

(0,4)

(0,5)

Radial Acoustic Modes(n=0)

(0,2)(0,3)(0,4)(0,5)

Higher-order modes (n>0) of an ideal sphere have a (2n+1)-fold degeneracy Real cavities will be asymmetrical,

therefore the originally degenerate modes will exhibit somewhat different resonance frequencies



Acoustic Resonances – Fluid-filled Shell

Solve for acoustic eigenvalues with shell motion completely accounted for using frequency dependant BC

0 5 10 150

10

20

30

40

50

60

70

80

Mode Number, n

Freq

uenc

y (k

Hz)

Frequency Spectrum

Acoustic resonance frequencies affected by resonances (extensional modes) of the shell

In detailed studies of gas-filled resonators, Moldover et. al. found that: Effect of radiation from shell to

surrounding fluid negligible (except near breathing resonance of shell)

Coupling of shell to mechanical supports minimal provided acoustic frequencies are high

Moldover, Mehl & Greenspan, JASA 79(2), 1986



Characterization of High Q Spherical Resonator

Kenneth BaderSESAPS Nov. 2007



Measured Shell Surface Vibrations

frequency (kHz)

Am

plitu

de (

dBm

)

30.3 kHz

49.3 kHz55 kHz



Set-up for LDV Measurements

Spectrum Analyzer

Power Amp SL Resonator LDV



LDV Preliminary Results LDV Preliminary Results



Set-up for Hydrophone MeasurementsSet-up for Hydrophone Measurements

Function Function GeneratorGenerator AmplifierAmplifier

ResonatorResonator

HydrophoneHydrophone

Automated Automated Positioning Positioning SystemSystem



Combined Pill, Hydrophone, and LDV Measurements



Power AmpLDV

Automated Automated Positioning Positioning SystemSystem

Pill

Function Generator

ScopeHydrophone

Characterization of High Q Spherical Resonators

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