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1 Oc679 Acoustical Oceanography Sonar Equation Parameters determined by the Medium Transmission Loss TL spreading absorption Reverberation LevelRL (directional, DI can’t improve behaviour) Ambient-Noise LevelNL (isotropic, DI improves behaviour) Parameters determined by the Equipment Source LevelSL Self-Noise LevelNL Receiver Directivity IndexDI Detector ThresholdDT (not independent) Parameters determined by the Target Target StrengthTS Target Source LevelSL

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2 terms are not very universal!

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3 Oc679 Acoustical Oceanography Units N = 10 log 10 where I 0 is a reference intensity the unit of N is deciBels so we might say that I and I 0 differ by N dB in terms of acoustic pressure, (p/p 0 ) 2 I/I 0 where the oceanographic standard is p 0 = 1 Pa in water we can write this in terms of pressure as 20 log 10 p/p 0 dB = 20log 10 p/p 0 1 0 √2 3 (double power level) 2 6 4 12 10 20 20 26 100 40 1000 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/p 0 =1/√2, I/I 0 = ½ dB = -3 we might say the -3dB level or ½ power level

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4 1 Pa is equivalent to 0 dB logarithmic scale linear scale dB

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5 Oc679 Acoustical Oceanography 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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6 light bulb source / implosive source roughly 6 db force ~ the pressure difference across the glass Energy available ~ δp 2 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

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8 Oc679 Acoustical Oceanography Pulsating Sphere – radial coordinatesmonopole instantaneous radial velocity at surface of pulsating sphere (radius a) is where U a 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 [ this is written differently than in text ] from this we can see that: 1.for constant U a, 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 (a 2 U a ) 3.for fixed k and a 2 U a, the radiated sound pressure is proportional to acoustic impedance A c CW source

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Dipole 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. Monopole 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 monopole movie dipole movie longitudinal quadrupole movie

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

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11 Oc679 Acoustical Oceanography Sound Sources including sources, the wave equation can be written as where the 3 terms on the RHS are mechanical sources of radiated acoustic pressure 1acceleration 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 2the divergence (spatial rate of change) of force (f) per unit volume – this is an adjustment to the conservation equation for momentum 3associated with turbulence as an acoustic source – particularly important in explaining the noise caused by turbulence of a jet aircraft exhaust 1 2 3

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12 Oc679 Acoustical Oceanography 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 dipole pressure expressed as 2 out-of-phase components for ranges large compared to separation l, use Fraunhofer approximation the small differences between R 1 and R 2 only important so far as defining phase differences (as kR 1, kR 2 ) – 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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13 Oc679 Acoustical Oceanography 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 R n and angle (that is, at Q): dimensionless amplitude factor of nth source rate of attenuation due to absorption + scattering the far-field approximation allows us to replace R n with R and we factor out the common terms leaving a factor that describes the transducers directional pressure response so D t has magnitude and direction

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14 Oc679 Acoustical Oceanography amplitude factors of the individual sources can be normalized such that the choice of a n 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, 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 boxcar triangle cosine Gaussian side lobe no side lobe main beam

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15 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 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

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16 Oc679 Acoustical Oceanography 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 D t is similar to a sinc function (but with a Bessel function involved) D t 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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17 Oc679 Acoustical Oceanography 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 R c beyond which interference is minimal ( that is, constructive interference cannot occur ) 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 P 0 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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18 Oc679 Acoustical Oceanography 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 transducer radius a half-intensity beam width defined at ½-power point: D 2 = 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 = 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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19 SOURCE LEVEL – recall our solutions to the acoustic wave equation acoustic impedance plane waves of form satisfy substitution into [w1] integrating w.r.t. x note resemblance to Ohm’s law V = ZI where V is voltage, Z is impedance and I is current A c, or rho-c is the acoustic impedance and is a property of the material property of the medium property of the wave

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20 atmospheric pressure (10 5 Pa = 10 11 Pa) typical source level ?

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21 c

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22 what about displacement? so the displacement required to produce a particular acoustic pressure, or a particular source level, is inversely proportional to frequency =

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24 example

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25 density microstructure

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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

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27 Oc679 Acoustical Oceanography % calibrate – all units are dB RL=20*log 10 (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*log 10 (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 range RL= 56 dB solve for: TS=RL-SL-RS+2.*TL = -70 dB Measured signal [V] / Sensitivity [V/ Pa] Example

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