Oc679 Acoustical Oceanography

1Oc679 Acoustical Oceanography

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

2

terms are not very universal!


Units

N = 10 log10 where I0 is a reference intensity

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

0

II

0

pp

p/p0 dB = 20log10 p/p0

1 0 √2 3 (double power level) 2 6 4 12 10 20 20 26 100 401000 60

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 = ½dB = -3we might say the -3dB level or ½ power level

4

1 Pa is equivalent to 0 dB

logarithmic scale

linea

r sca

le

dB


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

6

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

piezo – from Greek, “meaning to squeeze or press”

piezoelectric – generates a voltage when piezo’d (& v.v)

example: phonograph cartridge


Pulsating Sphere – radial coordinates monopole

instantaneous radial velocity at surface of pulsating sphere (radius a) iswhere Ua is the amplitude of the sphere’s radial velocity

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

i tr aR a

u U e

2 2A AR a a

ckp a U a UR 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

CW source

DipoleThis 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.

Monopole

QuadrupoleThis 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

monopole movie

dipole movie

longitudinal quadrupole movie

Relative radiation efficiency of a dipole: Relative radiation efficiency of a quadrupole:


Sound Sources

including sources, the wave equation can be written as 2 2

22 2 2

1 [ ( ) ( )]p mp f U U U Uc t t

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

1 2 3


The Dipole

2 equal out-of-phase monopoles with a small separation (c.f. ) between them

idealized dipole far-field directionality is a figure 8 pattern in polar coordinates (cos in polar coordinates)

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

1 20 0

1 2

1

0 0

exp( ) exp( )

1 11 cos , 2 1 cos2 2

exp( ) cos

d

d

d

p p p

i t kR i t kRp P RR R

R R R RR R

P Rp i t kR iklR

dipole pressure expressed as 2 out-of-phase components

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


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

pressure of nth source at range Rn and angle (that is, at Q):

1Wb

N

0 0 sinexp[ ( )]1

eRnn

n

a P R nkWp i t kR eR N

dimensionless amplitude factor of nth source

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

0 0

1

0

1

0

1

0

2 2 1/ 2

exp[ ( )] ,

sinexp( )1

sincos( )1

sinsin( )1

( )

eRt

N

t nn

t

N

nn

N

nn

t

P Rp D i t kR e R WR

nkWD a iN

D A iBnkWA a

NnkWB a

ND A B

so Dt has magnitude and direction


amplitude factors of the individual sources can be normalized such that

the choice of an determines the fundamental characteristics of the transducer

typically these choices are selected to 1. reduce side lobes2. narrow central lobe

one choice (the simplest) is ,

equal weighting to each transducer

other choices shown at rightthis 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

1

0 1Nnn a

1na

N

boxcar

triangle

cosine

Gaussian

side lobe

side lobe

side lobe

no side lobe

main beam

main beam

main beam

main beam


this can be extended to a continuous line source

sinc function, sin(x)/x which is the Fourier transform of a boxcar

far-field radiation from a boxcar line source

beam pattern in polar coordinates

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


Circular Piston Source far-field

this would be similar to the kinds of transducers we see in echosounders, ADCPsas a piston source this has uniform, in phase amplitude across a circular cross-sectionso 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 wavelengtha well-formed beam does not appear until ka becomes >> 1

we want the wavelength to be << physical dimension of the source transducer


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 )

2

c

aR

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


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

directional response of a circular piston transducerradius a

half-intensity beam width defined at ½-power point: D2 = 0.5 ( -3 db)

logarithmic polar plot ( ka = 20 )

One criterion is the half-intensity beam width

interpretive example:

consider 100 kHz source, a=10 cm

k = 2f/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

19

SOURCE LEVEL – recall our solutions to the acoustic wave equation

acoustic impedanceplane waves of form ( )i t kxu Ue satisfy

u uct x

substitution into [w1]

Ap ucx x

integrating w.r.t. x

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

( )Ap c u

property of the medium

property of the wave

20

atmospheric pressure (105 Pa = 1011 Pa)

typical source level ?

21

c

22

what about displacement?

A

dudt

p c

so the displacement required to produce a particularacoustic pressure, or a particular source level,is inversely proportional to frequency

=

23

24

example

25

density microstructure

26

acoustic backscatter strength

• dashed line – direct measurement calibrated using sonar equation

• solid line – model of microstructure backscatter [Bragg scattering from index of refraction fluctuations]

Use of SONAR EQUATION to calibrate microstructure backscatter


% 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;

this example taken from my code to calibrate BioSonics echosounder in units of dB

the 2 appears with TL here because the reflected signal is treated as a new source with the same range

TL= 29.3 dB at 35m rangeRL= 56 dBsolve for:

TS=RL-SL-RS+2.*TL

= -70 dBMeasured signal [V] / Sensitivity [V/Pa]

Example

Oc679 Acoustical Oceanography

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