Sound scattered by a body (Medwin & Clay Ch 7)
scattering is the consequence of the combined processes of reflection, refraction and
diffraction at surfaces marked by inhomogeneities in , c - these may be external or
internal to a scattering volume ( internal inhomogeneities important when considering
scattering from fish, for example )
net result of scattering is a redistribution of sound pressure in space – changes in both
direction and amplitude
for a monostatic system, we are most interested in the sound reflected back to the
source/receiver – this is termed backscatter
scattering is wavelength- (frequency, sort of) dependent
the sum total of scattering contributions from all scatterers is termed reverberation
this is heard as a long, slowly decaying quivering tonal blast following the ping of an
active sonar system
to start, we consider simple, hard, individual scatterers
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Oc679 Acoustical Oceanography
explosive source at 250 m
nearby receiver at 40 m
bottom depth 2000 m
reverb following explosive charge
initial surface reverb is sharp, followed by tail due to multiple reflection & scattering
then volume reverb in mid-water column (incl. deep scattering layer)
then bottom reflection, 2nd surface reflection, and long tail of bottom reverb
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transmission is a
gated sinusoid or ping
of duration tp
• sound pressure scattered by a small object – crests of sinusoid indicated as a sequence of wave fronts
• in this sketch ,  dependencies of incident wave are suppressed (this would be due to source beam
pattern)
• for simplicity wavefronts drawn as if coming from center of object – for a complex object, as shown,
there would be many interfering wave fronts spreading from the object
• within shadow, interference of incident and scattered waves is destructive as incident and scattered
waves arrive at the same time with same amplitude, but out-of-phase
• outside of shadow, interference of incident and scattered waves forms a penumbra (partial shadow)
• beyond penumbra ( < interfer), incident and scattered waves can be separated (no interference)
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traces of incident and scattered sound
pressures – ping has duration tp
travel time for sound to scatter to
receiver is R/c (here R is measured
from the scattering object)
this is referred to previous figure
incident sound pressure:
scattered sound pressure:
pinc (t )  Pinc e i 2 ft , 0  t  t p
pscat(t) = Pscat ei2πf(t-R/c)
, R / c  t  R / c  tp
= 0, otherwise
= 0, otherwise
Complex Acoustical Scattering Length, L(,,f )
consider amplitude and ignore phase
or
L( ,, f ) 
Pscat  Pinc L( ,, f ) 10R / 20 / R
Pscat ( ,, f )
R10R / 20
Pinc
dimension is length, unrelated to any
length scale of the body
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Differential Scattering Cross-section,  s ( ,, f )
2
Pscat
( ,, f )R 2 R /10
 s ( ,, f )  L( ,, f ) 
10
Pinc2
2
dimension area [ m2 ]
here, s, L, and Pscat all depend on the geometry of the measurement and the
carrier frequency, f, of the ping (really they depend on wavelength)
above is a bistatic representation in which source and receiver are at different
positions
when at same position (monostatic), it is called backscatter
now  = 0 and  = 0, and
2
 s (0,0, f )   bs (f )  L(0,0, f )
differential backscattering cross-section
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note that so far we have only considered the differential (and have used ) – that is we have
only considered a differential portion of the total radiated power from the scattering body –
this is done by looking at only a portion of the 3D surface with a finite receiver such as shown
in the schematic sketch
total scattering cross section
total scattering cross-section is the integral of  s ( ,, f )
over total solid angle
or
scat is the total power scattered by the body
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target strength
logarithmic measure of differential cross-section
  ( ,, f ) 
TS( ,, f )  10log  s 2
 [ db ]
1
m


reference area is 1 m2
  (f ) 
TS(f )  10log  bs 2  [ db ]
 1m 
for backscatter
 L( ,, f ) 
TS( ,, f )  20log 
 [ db ]
1
m


relative to scattering length
 L (f ) 
TS(f )  20log  bs  [ db ]
 1m 
or backscattering length
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how do we quantify a single transducer measurement
– R now referenced to transducer location
Pincei2πft = P0 ei2πf(t-R/c) R010-αR/20/R
Pscatei2πft = P0 ei2πf(t-R/c) R0Lbs(f) 10-2αR/20/R2
where P0 is referred to R0 (usually 1 m)
single source/receiver
travel time source to receiver is R/c
sound spreads spherically from source and then from object
pinc (t )  Pinc e i 2 ft , 0  t  t p
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Scattering by spheres
•
simple shape, well studied
•
an acoustically small and compact non-spherical body scatters in about the same way as a
sphere of same volume and same average physical characteristics (, c)
•
acoustically small  ( ka << 1 ) dimensions much less than those of incident sound
wavelength
1)
rigid sphere ka >> 1
reflection dominates, geometrical or specular scattering
2)
rigid sphere ka << 1
diffraction dominates, Rayleigh scattering
3)
fluid sphere – includes transmission through medium
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Oc679 Acoustical Oceanography
short wavelengths ka >> 1
or big objects
geometrical scatter from a rigid sphere ( ka >> 1 )
specular (mirrorlike) reflection
in the Kirchoff approximation ( discussed in text) plane waves reflect from an area as if the local,
curved surface is a plane
scatter consists of a spray of reflected waves each obeying simple reflection law – that is, angle of
reflection = angle of incidence
– we will employ a ray solution
diffraction effects are ignored - these would come from edge of shadow and behind sphere
we will calculate the scattering from a fixed, rigid, perfectly reflecting sphere at very high
frequencies ( ka >> 1 )
incident sound is a plane wave of intensity Iinc
no energy absorption in medium
no energy penetrates surface of sphere
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short wavelengths ka >> 1
or big objects
adi
dSicosi
2asini
a
i
dSi
di is a ring increment about the sphere
1st we need to know the incoming power at angle i
dSi  2 (a sin i )ad i
surface area increment (corresponding to grey
shaded area)
dS  dSi cos  i
component in direction of incident wave
d  inc  Iinc dS  Iinc 2 a 2 sin i cos  i d
input power to ring
or ,
d  inc  Iinc a 2 sin(2 i )d
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Oc679 Acoustical Oceanography
short wavelengths ka >> 1
or big objects
now, calculate scattered power
incident rays within angular
increment di at angle i are
scattered within increment
ds=2di at angle s=2i
the geometrically-scattered power measured at range R is
d  gs  Igs 2 (R sin s )Rd s
or
d  gs  Igs 2 R 2 sin2 i 2d i
assume all incident power is scattered [ no power loss ]
then d   d 
gs
and
inc
Igs
Pgs
a2
a

,
or

2
Iinc 4R
Pinc 2R
result:
scattered intensity is independent of
angle of incidence – which should be
the case by symmetry of the sphere,
but is not the case in general
in terms of scattering length
Lgs
Pgs

Pinc
R
where
Lgs
a

2
or
Lgs
a
2

1
2 
 0.28
[ we will get back to this later ]
in the case of the sphere, this means
that all differential geometrical
scattering cross-sections, including
backscattering cross-sections, are
equal
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Oc679 Acoustical Oceanography
short wavelengths ka >> 1
or big objects
the differential scattering cross-section is:
 gs  Lgs
2
a2
 , for ka>>1
4
and the total geometrical scattering cross-section is
 gs  4 gs   a 2 , for ka>>1 , where a2 is the cross-sectional area
gs does not include the effects of diffraction, so is not the total scattering cross-section
the ray solution is deceptively simple:
is accurate in the backscatter direction 0-90
but it ignores the complicated interference patterns beyond 90
more complete calculations using wave theory indicate that the total scattering cross-section
approaches twice its geometrical cross-section (2a2) for large ka
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Oc679 Acoustical Oceanography
long wavelengths ka << 1
or small objects
Rayleigh scatter from a sphere
ka << 1
when the wavelength is large compared to the sphere radius, scatter is due solely to diffraction.
2 simple conditions cause scatter:
1.
monopole radiation – in the case that the bulk elasticity (E1) of the sphere (recall E = pA/ =
compressibility-1) is less than that of water (E0), the incident condensations and rarefactions
compress and expand the body, thereby reradiating a spherical wave – phase reversed if E1>E0
2.
dipole radiation – if the sphere’s density (1) is much greater than that of the medium (0), the
body’s inertia will cause it to lag behind as the plane wave oscillates (sloshes back and forth).
This motion is equivalent to the water being at rest and the body being in oscillation. This motion
generates a dipole reradiation. When 1< 0, the effect is the same but the phase is reversed. In
general, when 1 0, the scattered pressure is proportional to cos, where  is the angle between
scattered and incident directions.
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Oc679 Acoustical Oceanography
long wavelengths ka << 1
or small objects
M&C develop a solution for the monopole and dipole radiation independently and then sum them
the scattered pressure is:
Pscat
 ikR
 2(ka)2  3
 a  Pinc e
 Pm  Pd  
 1  cos  
2
 2 R
 3 
scattering length and cross-sections determined by referencing R to 1 m
L(f , )  a
(ka )2  3

 1  cos  
3 
2

peaks, troughs at ka>1 due to interference
between diffracted wave around periphery
and wave reflected at front surface of sphere
2
(ka )4  3
 2
 s (f , ) 
 1  cos   a
9 
2

backscatter determined by setting  = 0
5(ka )2
Lbs  a
6
25(ka )4
 bs  a 2
36
relative scattering cross-section is obtained by
/a2 and is (ka)4
result: the acoustical scattering cross-section
for Rayleigh scatter is much less than for
geometrical scatter because sound waves bend
around and are almost unaffected by
acoustically small, non-resonant bodies
Rayleigh
scattering
geometrical
scattering
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Oc679 Acoustical Oceanography
fluid sphere, Rayleigh scattering
( ka << 1)
more general case, when sphere is an elastic fluid
2
e  1 g  1

 s (f , )  (ka) 

cos  a2, ka
 3e 2g  1

4
monopole component
1
dipole component
[ backscatter determined for  = 0 ]
•
•
•
•
g = 1/0, ratio of sphere’s density to
medium density
h = c1/c0
e = E1/E0
 = angle between incident and
scatter directions
most bodies in the sea have values of e and g close to unity and both terms are of similar
importance
bubbles
have e << 1 and g << 1
- in this case the monopole term dominates
- highly compressible bodies such as bubbles are capable of resonating when ka << 1
- resonant bubbles produce scattering cross-sections several orders of magnitude greater than
geometrical
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Oc679 Acoustical Oceanography
small target compared to λ
ka << 1
Rayleigh scattering
large target compared to λ
ka << 1
geometrical scattering
17
scattering of light follows essentially the same scattering laws as sound
but light wavelengths are much smaller than sound - O(100s of nm)
almost all scattering bodies in seawater are large compared to optical wavelengths and have
optical cross-sections equal to their geometrical cross-sections
 the sea is turbid to light
on the other hand, acoustic wavelengths are typically large compared to scattering bodies
found in seawater (at 300 kHz,  5 mm, 4 orders of magnitude larger)
- acoustic scattering is dominated by Rayleigh scattering
by comparison the sea is transparent to sound - what limits the propagation of 300 kHz
sound is not scattering but absorption
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Oc679 Acoustical Oceanography
why is the sky blue?
Rayleigh scattering α 1/λ4
λblue << λred
reds pass through atmosphere without scattering
but blues are scattered from O2 molecules and enters our eyes from a range of angles
violet scattered even better but our eyes are less sensitive to this
sunrise/sunset
at low azimuth, light passes through extended range of atmosphere
blue completely scattered out and sun appears red
green flash
occurs at the very end of the sunset (beginning of sunrise, when sun has passed below horizon
shorter wavelengths refracted more effectively than longer wavelengths
blue has been scattered out, reds are not effectively refracted
what’s left if green (sometimes very bright and pops up above horizon for < 1s)
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Oc679 Acoustical Oceanography
scattering from microstructure
hints of scattering from internal waves and microstructure in 60s, 70s
but confusing because of bio-scattering
alternatively, bio-scattering may be confused with microstructure
scattering leading to errors in estimating plankton populations
scattering cross-sections were computed based on Tatarski’s
computations for atmospheric radar (Proni & Apel 1975)
these were based on turbulence structure functions
note: these included the effects of velocity fluctuations as well as T (or c)
fluctuations
but uair/cair >> uwater/cwater
20
1st experimental evidence from controlled experiments in Wellington
reservoir, W. Australia (Thorpe & Brubaker, 1983)
“Observations of sound reflection by temperature microstructure” L&O
Known sources
Towed cylinder and weights at fixed depths from vessel 1
Measured using 102 kHz sounder from vessel 2
results
1. no signal when towed in mixed layer – thus
velocity fluctuations do not contribute
a-a, b-b, c-c are natural scatterers
2. clear signal of cylinder and weight wakes in
stratified regions
3. estimates of energy dissipated by towed
cylinder permitted estimates of turbulence
quantities, to which scattering theory could
be compared
21
theories employ the use of robust statistical models of turbulence spectra
Batchelor, 1957
(“wave scattering due to turbulence” Proc.Symp.Nav.Hydro.)
Goodman, 1990 (considers the bistatic or multistatic problem,
not just backscatter)
Ross etal 2004
22
Ross etal 2004
23
24
High-frequency acoustics – this is an important tool to help detect
instabilities that lead to turbulence
scattering from small-scale sound speed
sound speed c=c(T,p)
fluctuations caused by T and S microstructure
Ross and Lueck 2003
but here’s the problem
26
Andone Lavery WHOI
Lavery etal 2009
27
28
Lavery etal 2009
29
Lavery etal 2009
30
Lavery etal 2009
31
Lavery etal 2010
here all scattering is bio.
note difference in low k spectra
which tend to decrease toward low k
compared to turbulence spectra
32
33
34
35
coordinate systems
rectangular
cylindrical
spherical
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Oc679 Acoustical Oceanography
an object is effectively insonified by plane waves when its dimensions are
smaller than the 1st Fresnel zone – within the 1st Fresnel zone, a spherical
wavefront can be approximated as a plane wave
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Oc679 Acoustical Oceanography
definition of volume scattering strength (or
backscattering strength if referred to
same source and receiver
I
Sv  10log scat
Iinc
in term of surface scattering can define a
surface scattering strength
Ss  10log
Iscat
Iinc
Urick
note: the distinction in the definitions – the volume scattering strength Sv is defined by the ratio
Iscat/Iinc, each referenced to 1 m (or 1 yd) from the object
in M&C terms (that we have so far), Iscat is referenced to the receiver at range R from the object
while Iinc referred to 1 m from object– the inclusion of attenuation and spherical spreading (1/R2)
gives the length scale unit
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Oc679 Acoustical Oceanography
a more complete schematic of the problem includes beam pattern of single transducer as both
source (b) and receiver (b’)
( here absorption ignored )
• I0 is the axial intensity at unit distance (source level SL = 10 logI0)
• intensity at 1 m in direction (, ) is I0b(, )
• incident intensity at dV is I0b(, )/r2
• intensity backscattered at P 1 m back toward source is (I0b(, )/r2)SvdV
• scattered intensity at source is (I0b(, )/r4)SvdV, where it is assumed that sound spreads
spherically from both source and object dV
• receiver will produce voltage (rms) R2(I0b(, )b’(, )/r4)SvdV where R is the receiver
sensitivity
• total receiver output is  V[(R2I0SV /r4) b(, )b’(, )] dV
Urick fig 8.3
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