Sonar Equation
Parameters determined by the Medium
• Transmission Loss
TL
• spreading
• absorption
•Reverberation Level
RL (directional, DI can’t improve behaviour)
•Ambient-Noise Level
NL (isotropic, DI improves behaviour)
Parameters determined by the Equipment
• Source Level
SL
• Self-Noise Level
NL
• Receiver Directivity Index
DI
• Detector Threshold
DT (not independent)
Parameters determined by the Target
• Target Strength
TS
• Target Source Level
SL
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terms are not very universal!
2
Units
N = 10 log10 I where I0 is a reference intensity
I0
the unit of N is deciBels
so we might say that I and I0 differ by N dB
in terms of acoustic pressure, (p/p0)2  I/I0
where the oceanographic standard is p0 = 1 Pa in water
we can write this in terms of pressure as 20 log10
p/p0
dB = 20log10 p/p0
1
√2
2
4
10
20
100
1000
0
3 (double power level)
6
12
20
26
40
60
p
p0
for comparison:
• atmospheric pressure
is 100 kPa
• pressure increases at
the rate of 10 kPa per
meter of depth from the
surface down
compare p/p0=1/√2, I/I0 = ½
dB = -3
we might say the -3dB level or ½ power level
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dB
linear scale
1 Pa is equivalent to 0 dB
logarithmic scale
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Sources and Receivers
pulsating sphere– an idealization to ease theoretical treatment - this is a monopole source in
which the radiated sound is same amplitude and phase in all directions
[ bubbles act as natural pulsating spheres ]
sources and receivers (transducers) are actually designed with a wide range of properties
 physical, geometrical, acoustical and electrical
proper design is necessary to provide appropriate sensitivity to specified frequencies or to specific
propagation directions
skeleton of array used on subs
plane
transducer
array
cylindrical
transducer
array
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light bulb source / implosive source
roughly 6 db
force ~ the pressure difference across the glass
Energy available ~ δp2
A doubling of depth (100m to 200 m) ought to
result in a quadrupling of SL
this corresponds to 6 dB for a doubling in depth
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piezo – from Greek, “meaning to squeeze or press”
piezoelectric – generates a voltage when piezo’d (& v.v)
example: phonograph cartridge
Pulsating Sphere – radial coordinates
CW source
monopole
instantaneous radial velocity at surface of pulsating sphere (radius a) is ur
where Ua is the amplitude of the sphere’s radial velocity
R a
 Uaeit
combining this with the expression for the isotropic radiated pressure at R and the
particle velocity in radial coordinates gives an expression for the magnitude of the
acoustic pressure at R
pR   a2Ua 
 Ack
R
  a2Ua 
 A
R
[ this is written differently than in text ]
from this we can see that:
1.
for constant Ua, the radiated acoustic pressure is
proportional to the frequency of the CW source
2.
for a given frequency (written as either or k)
radiated acoustic pressure is proportional to the
volume flow rate at the source (a2Ua)
3.
for fixed k and a2Ua, the radiated sound pressure
is proportional to acoustic impedance Ac
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monopole
Monopole movie
Dipole
dipole movie
This can be considered as two equal strength monopoles that
are out of phase and a small distance, d apart (such that
kd<<1). There is no net introduction of fluid by a dipole. As one
source exhales, the other source inhales and the fluid
surrounding the dipole simply sloshes back and forth between
the sources. It is the net force on the fluid which causes energy
to be radiated in the form of sound waves.
Quadrupole
This can be considered as four monopoles with two
out of phase with the other two. They are either
arranged in a line with alternating phase or at the
vertices of a cube with opposite corners in phase. In
the case of the quadrupole, there is no net flux of
fluid and no net force on the fluid. It is the fluctuating
stress on the fluid that generates the sound waves.
However, since fluids don’t support shear stresses
well, quadrupoles are poor radiators of sound.
longitudinal quadrupole movie
Longitudinal Quadrupole
Relative radiation efficiency of a dipole:
Relative radiation efficiency of a quadrupole:
Sound Sources
including sources, the wave equation can be written as
1 2 p
 2m
 p  2 2   2   f   [  (U  )U  U ( U )]
c t
t
1
2
3
2
where the 3 terms on the RHS are mechanical sources of radiated acoustic
pressure
1
acceleration of mass per unit volume – this is associated with an
injection (or removal) of mass at a point on a sound source [ as for
example in a pulsating sphere or a siren (jet) which act as sources of
new mass ] – this appears in the mass conservation equation which is
later combined to get wave equation as it appears above
2
the divergence (spatial rate of change) of force (f) per unit volume – this
is an adjustment to the conservation equation for momentum
3
associated with turbulence as an acoustic source – particularly
important in explaining the noise caused by turbulence of a jet aircraft
exhaust
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The Dipole
2 equal out-of-phase monopoles with
a small separation (c.f. ) between
them
physically, it is straightforward to see
that the 2 out-of-phase signals will
completely cancel each other along a
plane perpendicular to the line joining
the 2 poles – they will partially cancel
everywhere else
 exp( i t  kR1 ) exp( i t  kR2 ) 
pd  P0R0 


R
R


1
2
pd 
1


R 1 
cos   ,
 2R

far-field directionality is a
figure 8 pattern in polar
coordinates (cos in polar
coordinates)
dipole pressure expressed as 2 out-of-phase components
pd  p  p
R1
idealized dipole
R2
1


R 1 
cos  
2R


P0R0
exp(i t  kR )ikl cos 
R
for ranges large compared to separation l, use Fraunhofer
approximation
the small differences between R1 and R2 only important so
far as defining phase differences (as kR1, kR2) – these are
combined using the dipole condition ( kl << 1 ) to get the
monopole pressure multiplied by iklcos
2 important results:
- radiated pressure reduced by small factor kl c.f. monopole
- radiation pattern no longer isotropic – directionality
proportional to cos, where  is the angle of the dipole axis –
max along dipole line and 0 perpendicular to line
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Line Arrays of Discrete Sources
transducers typically constructed of multiple elements – some general tendencies can
be discerned by consideration of a line of discrete sources equally separated over W
separation between elements is b 
W
N 1
pressure of nth source at range Rn and
angle  (that is, at Q):
pn 
anP0R0
nkW sin  R
exp[i (t  kR 
)]e
Rn
N 1
dimensionless
amplitude factor
of nth source
e
rate of attenuation due to
absorption + scattering
the far-field approximation allows us to replace Rn with R and we factor out the common
terms leaving a factor that describes the transducers directional pressure response
P0R0
exp[i (t  kR )]e  R , R
R
nkW sin
N 1
Dt   n 0 an exp(i
)
N 1
Dt  A  iB
p  Dt
e
nkW sin
)
N 1
nkW sin
N 1
B   n 0 an sin(
)
N 1
Dt  ( A2  B 2 )1/ 2
W
so Dt has magnitude and direction
A   n 0 an cos(
N 1
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amplitude factors of the individual sources can
be normalized such that

N 1
n 0
main
beam
an  1
the choice of an determines the fundamental
characteristics of the transducer
typically these choices are selected to
1. reduce side lobes
2. narrow central lobe
one choice (the simplest) is
an 
boxcar
side lobe
side lobe
triangle
main
beam
1,
N
equal weighting to each transducer
other choices shown at right
this is basically a plot of transducer gain as a
function of angle from the perpendicular
to the line array
tendency:
for the same number of elements,
weightings that decrease side lobes also
widen main beam
side lobe
cosine
main beam
Gaussian
main beam
no side lobe
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Oc679 Acoustical Oceanography
this can be extended to a continuous line source
sinc function, sin(x)/x which is the
Fourier transform of a boxcar
beam pattern in polar coordinates
far-field radiation from a boxcar line source
this is the result of a 1D line source (i.e., point sources aligned along a single coordinate axis)
suppose the sources aligned in 2D - this would result in a rectangular source whose directional
response would be the product of the response in the 2 coordinate directions
rectangular piston source
– individual elements are closely-spaced, in phase and have the same amplitude
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Circular Piston Source 
far-field
this would be similar to the kinds of transducers we see in echosounders, ADCPs
as a piston source this has uniform, in phase amplitude across a circular cross-section
so the directivity Dt is similar to a sinc function (but with a Bessel function involved)
Dt in polar coordinates – beam pattern of circular piston transducer
ka refers to the ratio of piston diameter to source wavelength
a well-formed beam does not appear until ka becomes >> 1
we want the wavelength to be << physical dimension of the source transducer
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Circular Piston Source

near-field
complex pattern in near-field due to
interference of radiation from different areas on
the disk
along disk axis, there is a critical range Rc
beyond which interference is minimal ( that is,
constructive interference cannot occur )
Rc 
a2

the range at which a receiver is safely in the
far-field is arbitrary, since some interference
will occur past the point where maximum
destructive interference ceases
this means that the definition of far-field in
part depends on what level of S/N is required
by the user
this complexity means that it is impossible to
measure P0 at 1 m from the source
rather it must be measured in the far-field and
extrapolated back to the source ( 1/R)
this is how SL is determined for use in the sonar
equation
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there are several descriptors that incorporate beam strength, radiation pattern and directivity
so that transducers can be compared quantitatively
as you find when selecting electronic components, different manufacturers use different criteria
- so you may need to do a little homework to understand what they mean – start with M&C sec
4.5.2 and other (better) transducer references but you may eventually have to talk to an
engineer
One criterion is the half-intensity beam width
directional response of a circular
piston transducer
radius a
logarithmic polar
plot ( ka = 20 )
half-intensity beam width defined at
½-power point: D2 = 0.5 ( -3 db)
interpretive example:
consider 100 kHz source, a=10 cm
k = 2f/c = 420 rad/m
[ this means the acoustic wavelength
is  = 2/k = 0.015 m ]
kasin = 1.6 when  = 2.2
half intensity beam width = 2 = 4.4
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SOURCE LEVEL – recall our solutions to the acoustic wave equation
acoustic impedance
i ( t  kx )
plane waves of form u  Ue
substitution into [w1]
satisfy
u
u
 c
t
x
p
u
  Ac
x
x
property of the medium
integrating w.r.t. x
p  (  Ac )u
property of the wave
note resemblance to Ohm’s law V = ZI where V is
voltage, Z is impedance and I is current
Ac, or rho-c is the acoustic impedance and is a
property of the material
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atmospheric pressure
(105 Pa = 1011 Pa)
typical source level ?
20
c
21
what about displacement?
d
= 
u
dt
p   Ac
so the displacement required to produce a particular
acoustic pressure, or a particular source level,
is inversely proportional to frequency
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23
example
24
density
microstructure
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Use of SONAR EQUATION
to calibrate microstructure
backscatter
acoustic backscatter strength
• dashed line – direct measurement
calibrated using sonar equation
• solid line – model of microstructure
backscatter [Bragg scattering from
index of refraction fluctuations]
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Example
% calibrate – all units are dB
RL=20*log10(bio.avsample);
% receiver level (this is what we measure)
SL=225;
% source level (from calibration)
RS=-39.25;
% receiver sensitivity (from BioSonics cal)
TL=20*log10(bio.depth-4.5)+0.045*(bio.depth-4.5);
% 1-way transmission loss at range bio.depth)
% solve for target strength [ RL=SL+RS-TL+TS ]
this is an additive function that replaces
the multiplicative system transfer function
TS=-SL+RL-RS+2.*TL;
the 2 appears with TL here because
the reflected signal is treated as a
new source with the same range
Measured signal [V] / Sensitivity [V/Pa]
TL= 29.3 dB at 35m range
RL= 56 dB
solve for:
TS=RL-SL-RS+2.*TL
= -70 dB
this example taken from my code to calibrate BioSonics
echosounder in units of dB
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