ORE 654
Applications of Ocean Acoustics
Lecture 5
Intense sounds: non-linear phenomena
Bruce Howe
Ocean and Resources Engineering
School of Ocean and Earth Science and Technology
University of Hawai’i at Manoa
Fall Semester 2011
3/24/2016
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1
Summary
• All previous discussion assumed infinitesimal
amplitude
• Extraordinary behavior when pressures large
– Steepening of wave slopes to produce shocks
– Streaming
– “parametric” sources – two high frequencies beat
to produce high intensity pencil beam
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Intense sounds:
non-linear phenomena
•
•
•
•
•
•
Harmonic distortion and shock waves
Cavitation
Parametric, difference frequency, sources
Acoustic radiation pressure
Acoustic streaming
Explosives as sound sources
• Any non-linear system – with high/finite
amplitude – harmonics and subharmonics; sum
and difference frequencies, from basic physics or
less-than-ideal systems
• Perturbations in, say density, no longer small,
need to include higher order terms
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3
Speed of sound
• Sound speed is function of
ambient pressure
• Hooke’s Law - linear Stress
(pressure change) and
strain (density change) slope determines c
• Bulk modulus of elasticity E
• Slope - sound speed - is a
function of ρA
• Slope and speed increase
with ρA
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pressure
density
p E
c 

  A
2
 p  1/2 
cA   


    
A
4
Equation of state
f ( p,V,T )  0
• To calculate c, need analytic relation
for equation of state
pV  nRT
• This form of equation (adiabatic – no
heat exchange) is well known for gases; p  p(  )
Γ is the ratio of specific heat at
constant pressure to specific heat at
constant volume; K depends on units
and atomic mass of gas
• For water, p includes not only external
pressure, but an internal cohesive
pressure of ~ 3000 atm. Γ and K not
ratio of specific heats but must be
determined empirically
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( p  )S  K

5
Include higher order term
• Differentiate to get sound
speed at ambient density,
and therefore c
• Expand density in Taylor
series and substitute
• Approximate by first two
terms of binomial
expansion
• Thus c reduces to cA if
density changes very small
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( p   )S  K;
p  K 
 p 
p
c   
 K  1

   s, 
2

 
  A  1 
 A 

c   K
 1 1/2
A

  
 cA 1 



A 
  
1 



A 
( 1)/2
( 1)/2
 (  1) 

; cA 1 
 ...
2 A

6
Alternative form for
equation of state
• Latter is empirical, can
try a different curve fit –
power series (Beyer,
1975)
• Equate with previous
(equivalent), slightly
different interpretation
• B/A – “parameter of
nonlinearity” used in
high intensity studies
• Water B/A = 5.0 at 20°C
and 4.6 at 10°C
• Air Γ = 1.4, B/A = 0.40
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B( /  A )2
p  A( /  A ) 
2
p A B
c2 

 2
  A
A
 (  1)  
c  cA 1 

2


A 
2
2



 (  1)  
2
2
c  cA 1  (  1)




A
4   A  


 
; cA2 1  (  1)



A 
A   A cA2  pA ; B  (  1) pA
B
  1
A
 B  
c  cA 1 

A
2


A 
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Steepening of waves
• Expression for u from
earlier (acoustic Mach
number definition et al.)
• Signal speed u + c
• Faster where pressure is
locally high and vice
versa

B  
c  cA 1 

 2A  A 

u
cA
A
 
B  
u  c  cA 
1
2A  

A
A

 
 cA  1  
 

A
B
  1
2A
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Harmonic distortion and
shock waves

p 
u  c  cA  1   
 

A
• Signal speed u+c
• Faster than cA where
pressure is locally high and
vice versa
• Different signal speeds at
different parts of wave
• Advance of crests relative
to troughs
• Sawtooth / repeated shock
• Loss of energy from
fundamental to harmonics
• Then high frequencies
dissipate due to absorption
Distance from source increasing
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Wave growth
• On axis particle travels
• Δx1 =cA Δt
• Peak of wave travels Δx2
in same time interval
• ΔX crest advance
relative to axis
• Assume distortedndwave is
fundamental + 2
harmonic
• The peak has zero slope
• For small kx (i.e., kΔX),
sin(kΔX) ≈ kΔX and
cos(kΔX) ≈ 1
• Relative strength of 2nd
harmonic P in terms of
fundamental P
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
 
x2  cA  1  
t
 

X  x2  x1  
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

cA t  
p  2P1 cos kx  2P2 sin 2kx
dp
0
dx
P1k sin kx ; 2P2 k cos 2kx
kX
P2 ; P1
2
 ( /  A )kXP1
P2 
; X  cA t
2


X
10
Wave growth
2nd harmonic pressure
• Use basic definition of c2 =
(Δp/Δρ) and say P1 ~ Δp
• For plane waves 2nd harmonic
growth ~
• square of fundamental P1
and
• number of wavelengths
progressed by fundamental
kX
• This for unattenuating plane
wave
• Spherical waves diverge, will
require greater initial amplitudes
to achieve same degree of
distortion
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P2 
 ( /  A )kXP1
2
; X  cA t
P1
  2
cA
 kXP12
P2 
2  A cA2
11
Wave growth - numbers
• For water B/A = 5, β = 3.5
• For air B/A = 0.4 and β = 1.2
• Strongest factor is denominator –
factor 16,000 larger for water
• Net effect – for same fundamental
pressure, 2nd harmonic grows grows
to same magnitude in a distance
1/5000 as far in air as in water, or
conversely, 5000 times as far in
water
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 kXP
P2 
2 c
2
1
2
A A
12
Wave growth – how far to formation of shock
wave?
2
• Say shock wave formed when 2nd
harmonic is half the fundamental
• Distance for this case?
• Sawtooth wave real situation; with
diverging spherical wave happens
at larger range
• Distance ~ 1/Mach, 1/non-linearity,
~ wavelength
• Must take into account absorption
(not here)
• Saturation limits sound energy that
can be input
• More energy – more harmonics –
more loss
• Higher intensity axial beam
attenuated more, beam broadens
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 kXP1
P2 
2  A cA2
1
P2  P1
2
 A cA2
X0 
 kP1
X0  (kM a  )1
M a  acoustical Mach number
u
P1


cA  A cA2
13
Wave growth saturation
• Saturation limits sound
energy that can be
input
• If linear, lines/curves
45°
• Lines at right
asymptotic limits
• For 10 m case, actual
level is about 6 dB less
than linear at ~550 kPa
(~235 dB re 1 μPa)
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Donald Ross
Cavitation
• In rarefaction/tension phase, pressure can
go “negative” and the medium ruptures
• Small bubbles always present near sea
surface are the nuclei for rupture initiation
• From Bernoulli effects/propeller blades
(mixture of dipole and monopole)
• Life processes (snapping shrimp)
• Increases chemical activity
• Erode metals, plastics, stones (kidney), …
• Light production – sonoluminescence
– Very high pressure, 30,000 K
– Picosecond light pulses
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Brian Pollack
15
Cavitation - 2
•
•
•
•
•
•
•
•
•
As sound levels rise, bubble resonance, harmonics generated
Bubbles generate subharmonics if driven near 2 x resonance
5% harmonic distortion for signals > 0.1 atm
If peak p > 1 atm (105 Pa = 220 dB re 1 μPa)
Negative pressure is trigger for sharp increase in distortion and
broadband noise, if CW f < 10 Hz
Function of f, duration, repetition, nuclei
If drive too hard, generate bubbles that decrease far-field sound
propagation
Gaseous cavitation - streaming bubble clouds jet away from
generation site (relative amounts of gas and water vapor ~
constant)
Vaporous cavitation – collapse of single bubbles - radiates shock
waves of broadband noise
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Cavitation - 3
• Nuclei often bubbles trapped in cracks/crevices of solid
particles
• Grow by “rectified” diffusion
– Start with small bubbles < 1 μm
– More gas diffuses into the bubble during expansion than out
during contraction when surface area smaller (more time is
spent large than small)
– At a critical radius, will grow explosively
• Threshold definition – distortion/harmonics and/or broad
band noise
• Above 10 kHz, steep increase in amount of CW pressure
amplitude to produce cavitation
• Large differences for “pure” water and tap or seawater
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Cavitation - 4
• Ocean-going transducer – regions on face
that exceed cavitation pressure limits,
combined with near surface bubbles
• “hot spots” – p > nominal
– Can have greater source levels at depth, high
ambient pressure effectively inhibits cavitation
– Fewer cavitaiton nuclei
– Smaller nuclei
– Streaming moves bubbles to new locations
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Cavitation
pulse duration and duty cycle
• Pulse duration < 100 ms, average acoustic intensity
required for 10 % distortion is >> than CW
• At low duty cycle can drive harder
cavitation
10%
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 Duty cycle
19
Parametric, difference frequency, sonars
• If two distinct intense sound beams are co-axial
at different frequencies, non-linearity creates
sum and difference frequencies
• Each beam modulates the other
• E.g., 500 and 600 kHz produce 100 kHz and
1100 kHz
• “parametric” or virtual sources distributed
along intense portion of interacting beams
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Parametric, difference frequency, sonars - 2
• Difference frequency – lower frequency (less
absorption)
• very narrow beam, ~same width as for the
primaries (but with smaller transducer)
• Acts as if a highly directional end-fire array
(“virtual end-fire array”)
• Bandwidth of difference frequency very
large
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Parametric, difference frequency, sonars - 3
• Two signals (ignore x
or observing location
kx = nπ)
• Instead of amplitude
at a point being
simple sum,
amplitude of p1 will
be modulated by p2
• Last term – nonlinear interaction,
strength m(P1,P2)
• Non-linear
interaction has
produced sum and
difference
frequencies
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p1 (t)  P1 cos( 1t);
p2 (t)  P2 cos( 2t)
P
P1  P1[1  m cos( 2t)]; m  2
P1
p(t)  P1 cos( 1t)  P2 cos( 2t)  P1m cos( 1t)cos( 2t)
2 cos x cos y  cos(x  y)  cos(x  y)
P1m
p(t)  P1 cos( 1t)  P2 cos( 2t) 
[cos(  t)  cos(  t)]
2
   1   2
   1   2
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Parametric, difference frequency, sonars - 4
• These sum and difference frequencies generated at all
points of intense interaction along beam
• Analogous to a line array of sources - end-fire
P1m
p(t)  P1 cos( 1t)  P2 cos( 2t) 
[cos(  t)  cos(  t)]
2
   1   2
   1   2
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Parametric, difference frequency, sonars - 5
•
•
•
•
•
•
•
•
More detailed analysis
Re-cast wave equation for secondary or “scattered” pressure, with non-linear source term
Westervelt’s wave equation for non-linear secondary tones
Assume two primaries with attenuation (they die off quickly)
Get get 2nd (and higher) harmonics and sum and differences
Difference frequency generated whenever P1P2 large, contribution from beam near
source largest
Once generated – launched, “on its own” (Huygen’s wavelets along beam)
Primary and sum frequencies die off
 2 ps 1  2 ps

 2
 2
2
2
x
c t
c
 1  2 ( p1  p2 )2 
 c2

2
t


...
 d2 P1P2 S0
pd (R,  ) 
exp( d R)cos( d  kd R   )
4
4 Rc
4 

   e2  kd2 sin 

2
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 kd
2 
 OREarctan
sin 
654 L5 
2
 e
1/2
24
Parametric, difference frequency, sonars – 6
• Narrow beam pattern (high directional resolution with small
transducer)
• Beamwidth relatively insensitive to changes in difference frequency
• No side lobes in Dd (secure acoustic comms)
• Inherent broad bandwidth
1/2
2
 k 

4 
• Projector cavitation not a problem
d
Dd ( )  1    sin 
2
   e 

 e 
sin   
2  kd 
d
e
kd
1/2
 1
 e 
2d ; 4  
k 
1/2
d
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Parametric, difference frequency, sonars - 7
•
•
•
•
•
Increase in bandwidth
BW of a primary typically ±5%
f1 = 418 ± 21 kHz; f2 = 482 ± 24 kHz
fave = 450 ± 22.5 kHz
fdiff = 64 kHz ± 22.5 kHz - ±35%
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Parametric source
example f  418 kHz;418f482482 kHz
1
2
average f 
• Given 2 f’s, what is
beam width?
• What size piston
to produce the LF?
• What size piston
needed?
• What is the
reduction in
source radius for
same beamwidth?
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 450 kHz;
ave  0.33 cm
2
difference f  482  418  64 kHz; d  2.3 cm
 e (450 kHz) = 0.15 dB/m =1.7  10 -2 Np/m
2
2
1
kd 
0.023
; 268 m ; kave 
 
2d ; 4  e 
 kd 
1/2
0.0033
 1.7  10 
 4
 268 
-2
; 1855m 1
1/2
 0.032 rad = 1.8o
for circular piston kd a  sin d  1.6
1.6
1
ad  1.6(kd sin d ) 
 38 cm
o
268 sin 0.9
1.6
1
aave  1.6(kave sin ave ) 
 2.5 cm
o
1855 sin 2.0
aave
 0.065
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ad
Parametric source
efficiency
W W
1
2
P1  P2
• Same power W
and pressure P
• Power radiated
through
primary beam
area S0
• On-axis rms Pd
(using beam
width)
• Average
intensity ~ P2d
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S0 P12
1   2 
c
 d20
Pd 
2 8 Rc 3 e
2  fd  0
Pd 
2Rc 2d2
 d  average intensity (axial, ~Pd2 / c)
 beam area  (Rd )2
 2 fd2 20
d 
(2 c 5d2 )
 d  2 fd2  0


0
2 c 5d2
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Parametric source
example
• For preceding parameters, P1=100W
• Efficiency = 0.7%
• Increase efficiency:
– Increase difference frequency
– Increase power
– Decrease beamwidth (lower primary
frequency with constant beam area –
larger transducer)
• Only Power increase without
sacrificing advantages – limit by
saturation effects and beam
broadening and cavitation
• Level can be increased by non-linear
oscillation of bubbles, but some loss
in radiation directionality
• Used for sub-bottom profiling
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d  f 


0
2 c 
2
2
d
0
5 2
d
29
Tritech SeaKing Parametric SBP SubBottom Profiler
•
•
•
•
•
•
•
•
•
•
•
•
Primary frequency 200 kHz
Primary beamwidth 4 degrees
Low frequency 20 kHz
Low frequency beamwidth 4.5 degrees
Pulse length 100 μseconds
Range resolution of HF Dependent upon rangescale (10-100mm)
Range resolution of LF Dependent upon rangescale (60 μseconds@30m)
Power requirements 24VDC @ 410mA (Nominal for DST model)
Transducer 200 mm diameter
Weight in air 6.3kg
Weight in water 2.7kg
Maximum operational depth 4000m
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Acoustic streaming
• Non-linear – harmonic
distortion and shocks
• Can also cause
unidirectional flow
– “quartz wind” outward
jetting or drift of water
in front of transducer
– Strongest on axis,
distances of meters
– Can be >1000 acoustic
velocity
– Eddy/recirculate/3-D
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Movie -1
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Movie - 2
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33
Acoustic streaming –
radiation pressure
• Langevin radiation
pressure = average
momentum carried
through unit area in unit
time = time average of
momentum per unit
volume ρAu by particle
velocity u (U = rms
velocity)
• This also = average
energy density in beam =
<ε> = average intensity /
cA
3/24/2016
momentum
 Au 2  AU 2
area  time
p 2 / cA
I
P2
 

 2  AU 2
cA
cA
cA
PRL 
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Acoustic streaming –
momentum transfer
• Spatial change in momentum – absorption
/ dissipation
• Newton’s 2nd law – rate of change of
momentum per unit area is force per unit
area or change in pressure ΔPA across slab
dx
• Loss of acoustic pressure in path dx
creates pressure gradient causing flow
• Or, loss of momentum in acoustic beam
made up with gain in momentum of fluid
mass – conservation of momentum
• Bubbles on a transducer face – cavitation,
asymmetric toroid produces destructive
jet into wall, but superimposed on mean
flow pattern
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I 2 e I x

; I  2 e I x
cA
cA
change of momentum
area  time
2 e I x
PA 
cA
PA 
35
Acoustic streaming
velocity u2
• Non-linear – magnitude
proportional to intensity – Eckart,
1948
• For ideal beam in a tube P(r) = P for
r<a and P(r) = 0 for a0≥r≥a
• a = radius of non-divergent sound
beam
• a0 (larger) radius of tube
• Measurements -> calculate bulk
viscosity μ’ from intensity and
streaming velocity
• Liebermann (1949) helped resolve
difference between theoretical and
experimental values of attenuation
• For fluids in general
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2 2 f 2 a 2GIb
u2 (0) 
cA4
G  (a 2 / a02  1) / 2  log(a / a0 )
4
b    '/ 
3
I  P 2 / ( cA )
 '  dynamic bulk viscosity
  dynamic shear viscosity
36
Explosives as sound sources
• TNT etc ~ 4,400 J/g = 1050 cal/g
– Rapid reaction/detonation – 3000 °C, p ~ 50,000
atm
– Detonation velocity 5,000 - 10,000 m/s
• Gunpowder – burning – 0.3 m/s, slowly growing
• Two sources of sound
– The shock wave ~ half the energy, propagates at > cA
– Large oscillating gas bubble / gas globe
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Explosives as sound sources
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Shock wave
• Instantaneous rise in
pressure Pm
• Then exp decay, time
constant τs s
• Both scale by (w1/3/R)
– w weight of
explosive kg, R range
m
• Common SUS 0.82 kg
• No attenuation
exponents 1 and 0
• Absorption and nonlinearity
3/24/2016
PM  50.94(w
1/ 3
1.13
/ R)
MPa
p  pm exp(t /  s )
 s  8.12  10 5 w1/ 3 (w1/ 3 / R)0.14
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Shock front propagation; the
Rankine-Hugoniot equations
• Earlier wave equation –
infintesimal waves
• Conservation of mass
• Conservation of
momentum
• Water only slightly
compressible so density
ratio ~ 1
• Shock speed U depends
on average slope dp/dρ
in p(ρ)
• Speed of sound of peak
cm depends on local
incremental
slope
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M   AU  m (U  um )
 m   A 
um  
U

 

m
pm  pA   AU 2  m (U  um )2
pm  pA   AumU
( pm  p A )  m
U
(  m   A ) A
p
c 

2
m
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pm ,  m
cm  U  c A
40
Shock front velocity
( pm  pA )m
U
(  m   A ) A
• Need equation of
state and
conservation of
momentum
• Speed of shock u + c
can be > c
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Gas globe
• Contains chemical gas products and water vapor
• Contains half total energy of explosion
• Initial acceleration, expands, decelerates,
continues past radius where internal p equals
external, reaches maximum radius with internal p
less than external, bubble contracts, oscillates,
produces bubble pulses
• Period of oscillation f(energy after shock,
ambient pressure and density)
• Assume spherical, need partition of non-shock
energy, Y
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Gas globe - frequency
• Assume spherical, need partition of
non-shock energy, Y (Joules)
• At maximum radius am, KE is zero,
internal energy << PE
• Assume all non-shock energy (~1/2
explosion energy) Y is PE
• Period of spherical bubble in ambient
(future derivation)
• Substitute
• Period T ~ depth and yield
• Real –
– not spherical (large ambient p
difference)
– often splits in two because of dimpling
– Non-sinusoidal oscillation
– Frequency changes as bubble rises
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4 3
Y   am p A
3
 3Y 
am  
 4 pA 
T  2 am
1/ 3
A
3YpA
5 /6
T  KY 1/ 3 1/2
p
; K; 2
A
A
43
Interaction with
ocean surface
• Reflections off surface,
phase reversed
• “noise” after reflection
– under
tension/negative
pressure – cavitation,
microbubbles radiate,
causes reflected shock
to loose energy
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44
Perth-Bermuda 1960
•
•
•
•
•
•
•
•
•
•
300 lb TNT (4400 J/g)
1000 m deep
w = 136 kg
R=1m
Pm = 318 MPa
τ = 0.0002 s
PA = 107 Pa (1000 m)
Y = 3x108 J
am = 1.9 m
T = 0.06 s
4 3
Y   am p A
3
PM  50.94(w1/ 3 / R)1.13
MPa
 s  8.12  10 5 w1/ 3 (w1/ 3 / R)0.14
 3Y 
am  
 4 pA 
1/ 3
5 /6
T  KY 1/ 3 1/2
p
; K; 2
A
A
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45
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Explosion test facility
• Brett et al., An
experimental facility for
imaging of medium scale
underwater explosions,
DSTO-TR-1432, 2003
• Defense Science and
Technology Organization,
Australia
• Near Melbourne
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46
Video
• 0.5 kg
• 1 ms resolution
• Frame 4 – shock wave
causes cavitation on
camera window at 4 ms
• Initial phase spherical and
smooth (note sunlight)
• During contraction –
asymmetric, flattened base
– pA(z)
• Bubble rises about 0.4 m
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P(t)
•
•
•
•
•
•
•
•
•
Rapid expansion and collapse near
minimum radius
Slow at maximum radius
Rmax – 1.14 m, rmin – 0.27 m in 97 ms
6.1 m3 in 0.09 s – 6 tons
68 m3/s
Vmax = 3.6 m/s
Hydrophone 4.5 m range
Shock wave, bubble pulse at min radius
Surface reflect at 4.5 ms, walls/floor 13-19
ms
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Acoustic radiation
pressure
• Sound beam transports acoustic
energy and momentum
• Static pressure difference between
inside and outside pressure <ε> =
Langevin radiation pressure
• (Newton – rate of change of
momentum = force)
• Momentum per unit volume x
velocity
• At wall velocity = 0
• Average energy density in beam =
momentum transfer
<ε> = average intensity / cA
PRL 
 Au 2  AU 2
area  time
• Diaphragm inserted in beam and
radiation pressure measured – static
2
p
/ cA
I
P2
displacement
 

 2  AU 2
cA
cA
cA
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51