ORE 654Applications of Ocean Acoustics

Lecture 2Sound propagation in a simplified sea

Bruce HoweOcean and Resources EngineeringSchool of Ocean and Earth Science and TechnologyUniversity of Hawaii at ManoaFall Semester 2011**ORE 654 L2

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Sound propagation in a simplified seaSpeed of soundPulse wave reflection, refraction, and diffractionSinusoidal, spherical waves in space and timeWave interference, effects and approximations1-D wave equationPlane wave reflection and refraction at a plane interface3-D wave equation*ORE 654 L2*

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Speed of sound - FirstColladon and Sturm (1827)Lake Geneva1437 m/s at 8 C

Sea water speed is greater*ORE 654 L2*

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Speed of sound - SeawaterSound speed (c or C m/s) is a complicated function of temperature T C, salinity S PSU, and pressure/depth z mSimple formula by Medwin (1975): c = 1449.2 + 4.6T 0.055T2 + 0.00029T3 + (1.34 0.010T)(S 35) + 0.016zOthers: Mackenzie, Wilson, Del Grosso, and Chen-Millero-Li newest TEOS-10Note: for deep ocean, uncertainty is likely 0.1 m/s at depth, still ?

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Speed of sound gravity affects pressureTo convert pressure p (dbar) to depth z (m) use Saunders, 1981Accounts for variation of gravity with latitude z = (1 c1)p c2p2c1 = (5.92 + 5.25 sin2) 103, latitude c2 = 2.21 106Assumes T = 0 C and salinity 35 PSUAdditional dynamic height correction available if necessary*ORE 654 L2*

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Speed of sound, range, travel timeC = R/TR = C/TT = R/C

PerturbationsIncrease in range increases travel timeIncrease in sound speed decreases travel time

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Speed of sound measuringSound velocimetersNeeded forNavigationSonarsMeasure the ocean temperatureInverted echosoundersTomographyNot so easyTime and distance accuracy1 part in 104 best

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Speed of sound - Seawaterc = 1449.2 + 4.6T 0.055T2 + 0.00029T3 + (1.34 0.010T)(S 35) + 0.016z

Differentiate gives C 4.6 T + 1.34 SSo T = 1 C 5 m/s in sound speedAnd 1 PSU 1 m/sIn practice temperature variations are large and far out weight salinity variations (which are typically small)

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Pulse wave propagationTiny sphere expandingHigher density condensationImpulse/pulse moves outwardLongitudinal wave displacements along direction of wave propagation*ORE 654 L2*

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Acoustic intensityFluctuating energy per unit time (power) passing through a unit areaJoules per second per meter squaredJ s-1 m-2 = W / m-2 Conservation of energy through spherical surface 1 and through surface 2Sound intensity (~ p2) decreases as 1/R2Total pulse energy would be integral over time and sphere*ORE 654 L2*

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Huygens Principle*ORE 654 L2*Qualitative description of wave propagationPoints on a become wavefronts bWavelet strength depends on direction Stokes obliquity factor

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Reflection*ORE 654 L2*Successive positions of the incident pulse wave at equal time intervals (R=ct) over a half spaceSuccessive positions of reflected pulse wave frontsReflection appears to come from image of sourceLaw of reflection: 1 = 2

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Snells Law of Refraction*ORE 654 L2*equal travel time R/C

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Fermats PrincipleEnergy/particles can and do take all possible paths from one point to another, but paths with the highest probability (in our case) are stationary paths, i.e., small perturbations dont change them.In practice, these are paths of minimum travel time principle of least time.*ORE 654 L2*

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Snells Law and Fermats Principle*ORE 654 L2*Travel timeDifferentiate and set to zero to find minimumP is minimum travel time pathPPA

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DiffractionIncident, reflected, and diffracted wave frontsDiffracted portion fills in shadowsAll three = scattered sound redirected after interaction with a body

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Sinusoidal, spherical waves in space and time(a) pressure at some time(b) range dependent pressure at some instant of time(c) time dependent pressure at a point in space*ORE 654 L2*

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SinusoidsSpatial dependence at large range, pressure ~ 1/RTime and spaceRepeat every 2 or 360Period T=1/f

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Sinusoids - 2Radially propagating wave having speed cPick an arbitrary phase at some (t,R). At later t+t, same phase will be R+RWith negative sign waves traveling in positive directionWith Positive sign, negative direction

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Wave interference, effects and approximationsConstructive and destructive interference from multiple sourcesAdd algebraically for linear acoustics, not so for non-linearApproximations are useful tools

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Local plane wave approximationAt a large distance from sourceIf restrict /8 (45)Then W (R)1/2 *ORE 654 L2*

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Fresnel and Fraunhofer approximationsAdding signals due to several sinusoidal point sourcesSeparate temporal and spatial dependenceFraunhofer long rangeFresnel nearer rangesConvert differences in range to phase - decide

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Near field and far field approximationsNear field differential distances to source elements produce interferenceFar field beyond interference effectsCritical range*ORE 654 L2*

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Interference between distant sources:use of complex exponentials

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Interference between distant sources:use of complex exponentials - 2*ORE 654 L2*Maximum value is 4P2 and minimum is 0Interference maxima at k(R2-R1) = 0, 2, 4, and minima at , 3, 5, Cause pressure amplitude swings between 0 and 2

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Point source interference near the ocean surface: Lloyds mirror effect*ORE 654 L2*Sinusoidal point source near ocean surface produced acoustic field with strong interference between direct and reflected soundAbove surface imageFunction of frequency and geometryPressure doubling in near region, Beyond last peak pressure decays as 1/R2 (vs 1/R)

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1-D wave equationNewtons Law for AcousticsConservation of Mass for AcousticsEquation of state for acousticsCombine to get wave equationSmall perturbations in pressure and density around ambient*ORE 654 L2*

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Newtons Law for AcousticsPoint source, large R, plane waveLagrangian frameNet pressureMultiply by area to get net forceMass is density x volumeAcceleration is du/dtF = ma*ORE 654 L2*

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Conservation of mass for acousticsEularian frameNet Mass flux into volume is difference (over x) between flux in and out where flux is density x velocity x volume elementThis must balance rate of increase in mass increase

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Equation of state for acousticsRelation between stress and strainHookes Law for an elastic body: stress ~ strainFor acoustics, stress (force/area) = pressureStrain (relative change in dimension) = relative change in density /AProportionality constant is the ambient bulk modulus of elasticity EHolds for all fluids except for intense soundAssumes instantaneous P causes instantaneous (time lag molecular relaxation absorption)*ORE 654 L2*

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Wave equationPartial x of F=maPartial t of conservation of massCombineUse equation of state to replace density with pressureDefine sound speedFinal standard form equation*ORE 654 L2*

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ImpedanceRelate acoustic particle velocity to pressure in a plane wave (general form of wave equation solution)General solution +/-Wave traveling in +x has velocitySubstitute into F=maIntegrate over xAnalogous to Ohms LawPressure ~ voltageVelocity ~ currentSpecific acoustic impedance Ac ~ electrical impedance

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Mach numberIn fluid mechanics dimensionless numbers are often very usefulRatio of acoustic particle velocity to speed of sound Take plane wave and conservation of massM measure of strength and non-linearity*ORE 654 L2*

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Acoustic pressure and densityUse impedance and Mach number relationsLiquids, equation of state p=p() is complicated so inverse used eqn of state calculated from accurate measurements of sound speed *ORE 654 L2*

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Acoustic intensityIntensity (vector) = Flux = (energy / second = power) perpendicular though an area J/s m-2 = W/m-2Remember Power = force x velocityIntensity = (force/area) x velocityElectrical analogPower = voltage2 / impedanceIf sinusoid, use rms = 0.707 amplitude

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Plane wave reflection and refraction at a plane interfaceDerive reflection and transmission coefficientsApplicable for spherical waves at large range i.e., waves are locally plane*ORE 654 L2*

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Reflection and transmission coefficients - 1Use physical boundary conditions at the interface between two fluidsBC-1: equality of pressure BC-2: equality of normal velocity

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Reflection and transmission coefficients - 2Velocity BCAngles by Snells Law*ORE 654 L2*

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Reflection and transmission coefficients - 3Pressure BCAll time dependencies at the interface the sameReflection and transmission coefficients

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Reflection and transmission coefficients - 4Pressure BCTake pi as reference, divide through by it

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Reflection and transmission coefficients - 5Velocity BC

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Reflection and transmission coefficients - 62 equations, Solving for R and TConnected by Snells Law

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Reflection and transmission at surface -1Important at surface and bottomSurfacewater = 1000 kg/m3 >> air = 1 kg/m3cwater = 1500 m/s > cair = 330 m/swatercwater >> aircair (~3600)Take 0 water to airR -1 and T 4 10-4pr = -pi so near zero total pressure at interface but ur = 2ui so particle velocity doublesWater to air interface is a pressure release or soft surface for underwater soundWater to air extreme case of c2

Reflection and transmission at surface -2From air to waterPressure doubling interfaceZero particle velocityFrom air, surface is hard

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Reflection and transmission at bottom - 1From ocean to bottom cbottom > cwater c2 > c1 Possibility of total internal reflectioni > c critical anglec = arcsin(c1/c2)If i > c rewrite Snells Law

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Reflection and transmission at bottom - 2Angle of incidence > criticalSnells Law becomesExp decay into medium 2, skin depth z

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Plane wave reflection at a sedimentary bottomShallow water south of Long IslandAssume sediments are fluidR12 bottom loss BL = -20 log10 R12 thin layers one composite layer; thickness< other distance scales*ORE 654 L2*

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Plane wave reflection beyond critical angleCan have perfect reflection with phase shiftUseful: virtual, displaced pressure release surface (R12 = -1)Virtual reflector*ORE 654 L2*

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Spherical waves beyond critical angle: head waves - 1When incident wave is at critical angle, a head wave is producedMoves at c2, radiates into source medium c1Travels at high(er) speed, arrives firstAppears to be continually shed into slower medium at the critical angle sinc = c1/c2Fermats Principle Minimum travel time*ORE 654 L2*

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Spherical waves beyond critical angle: head waves - 2More detailed analytical development yields amplitudeAlso for source under ice plates 100s m, 1-2 m thickModel latter scale lengths and properties 3.3 mm acrylic at 62 kHz = 1 m thick ice at 200 Hz; critical angle 39thin ice covered by air NOT =simple water-air pressure release interface*ORE 654 L2*

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Spherical wave reflection from a finite reflector: Fresnel zones - 1Reflection form a circular planeCircular rings Fresnel zonesRadii such that differencemagnitude of reflection = f(, h, r, R12)

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Spherical wave reflection from a finite reflector: Fresnel zones - 2*ORE 654 L2*Different rings, different distances from sourceCan be cancellation or increased signalFinite disk, sum over all elements dSPhase of wavefront traveling distance 2RInterested in phase change 2kR; smallest (reference) value is 2khRelative phase difference Solve for R and then r as function of Interest in large separation (first term in r2 only)First phase zone (central white circle) 0 positiveIn next (first dark ring) phase is - 2 - negative

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Spherical wave reflection from a finite reflector: Fresnel zones - 3Formula for radii of the n ringsn = 1, reflected signal 2x pressure as infinite planefor n = 2, reflected ~ 0For infinite plate (r = ), Pr equivalent to virtual image h behind disk/reflector, factor 1/2h pressure inversely proportional to range for spherical divergence

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3-D wave equation - 11-D plane wave not adequate in many casesShallow water cylindricalFish cylindersScattering by spheres spherical (or expand in terms of plane waves)General equation divergence of the gradient of p = Laplacian*ORE 654 L2*

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3-D wave equation - 2Laplacian in 3 coordinate systems*ORE 654 L2*

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Continuous waves in rectangular coordinatesUse separation of variablesEach term function of only one variable, so each of terms must = one constant (factor of c2 between space and time)

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Continuous waves in rectangular coordinatesTry exponential forms

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Continuous waves in rectangular coordinatesSubstituting in first equationPlane wave in +x, +y, +z

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Omnidirectional continuous waves in spherical coordinatesAssume spherical symmetry so no angular dependence dependence only on R and t (e.g., point source)Solution analogous to 1-D rectangular with p replaced with Rp and x by R = - outgoing, + incomingP0 usually at unit distance R0

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Acoustic pressure for sinusoidal, omnidirectional wavesAgain, separate variables radial and temporal functions

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Particle velocitycontinuous wavesStart with acoustic force equation radial componentClose to source, small kR, quadrature component which lags pressure by 90Explosion motion lags pressure pulseLarge kR like plane wave u ~ p

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Far field intensityFrom earlier, conservation of energy showed intensity proportional to 1/R2For kR >> 1, particle velocity ~ pNow using functional dependence (ct-R)At long range iR is simple product of p and uRIn the far field of a point source, sound pressure and velocity decrease as 1/R and intensity as 1/R2Now shown from first principles, wave equation

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Sound propagation in a simplified seaSpeed of soundPulse wave reflection, refraction, and diffractionSinusoidal, spherical waves in space and timeWave interference, effects and approximations1-D wave equationPlane wave reflection and refraction at a plane interface3-D wave equation*ORE 654 L2*

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Next week8/30 Tuesday Transmission and attenuation along ray pathsEnergy transmission in ocean acousticsRay paths and ray tubesRay paths in refracting mediaAttenuationSONAR equationDoppler shifts*ORE 654 L2*

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Sound is a traveling mechanical disturbance. Here in a fluid. Many ways to characterize primarily with pressure, then velocity and density.

Assume medium is homogeneous (same physical properties at all points)Isotropic (same propagation properties in all directions)No absorption (no conversion to heat)No dispersion (no dependence of sound speed on frequency)Very small acoustic pressure (or other) linearPhenomena:Scattering diffraction (bending), scattered back reflection, refraction at interfacesMultiple waves interference

Use these to learn about the ocean

We treat simple cases here, simplifying assumptions.

*Sound is a traveling mechanical disturbance. Here in a fluid. Many ways to characterize primarily with pressure, then velocity and density.

Assume medium is homogeneous (same physical properties at all points)Isotropic (same propagation properties in all directions)No absorption (no conversion to heat)No dispersion (no dependence of sound speed on frequency)Very small acoustic pressure (or other) linearPhenomena:Scattering diffraction (bending), scattered back reflection, refraction at interfacesMultiple waves interference

Use these to learn about the ocean

We treat simple cases here, simplifying assumptions.

*Fig. 1.1. Colladon and Sturms apparatus for measuring the speed of sound in water. A bell suspended from a boat was struck under water by means of a lever m, which at the same moment caused the candle l to ignite powder p and set off a flash of light. An observer in a second boat used a listening tube to measure the time elapsed between the flash of light and the sound of the bell. The excellent results were published in both the French and German technical literature.(Annales de Chimie et de la Physique 36, [2], 236 [1827] and Poggendorffs Annalen der Physik und Chemie 12, 171 [1828].)*Fig 1.2 Spherical spreading of a pulse wavefront. The instantaneous intensity is i0 at the radius R0 and later is iR at radius R.*Later will show sound intensity is a fucntion of pressure squared.Could have been an implosion then rarefaction pulse*Fig. 1.3. Huygens wavelet construction for a pulse. (a) Points on a previous pulse wave front at a are the sources of wavelets whose envelope becomes the new wave front, b. The wave front thereby moves from a to b. (b) The dependence of wavelet strength on propagation direction, is shown by shadowing. The analytical description the dependence on , (1.4) is called the Stokes obliquity factor.*Fig 1.4. A spherical pulse from a point source and its reflection at a rigid, plane reflector. The usual penetration of the pulse into the lower half space behind the plane face of the reflector is not shown. (a) Successive positions of the incident pulse wave over a half space. (b) Huygens constructions of successive positions of the reflected pulse wave fronts. The reflected pulse wavefront appears to come from and image source in the lower half space. A homogeneous, isotropic medium is assumed. The geometry shows 2 = 1Lower half is image space. Rays are perpendicular to the wavefront. Here, reflection is assumed perfect (R=1).

Reflection appears to be expanding as if it came from a source beneath the reflecting surface.

*Fig. 1.5. Huygens construction for a compressional plane pulse at a sequence of times and positions in adjacent media showing Snells Law of Refraction.

Assume very distant point source so wavefronts are effectively planar (wavefront curvature nil)

Here c2>c1 (air to water, or deep ocean or where pressure is the only effect governing sound speed)

Ignore any reflection (R=0)

Sometimes given in terms of grazing angle replace sin with cos

*A result, not a cause

The modern version of Fermat's principle states that the optical path length must be stationary, which means that it can be either minimal, maximal or a point of inflection (a saddle point). Minima occur when a wave passes from one medium into another (refraction) and in the reflection of light from a planar mirror. Maxima occur in gravitational lensing. A point of inflection describes the path light takes when it is reflected off an elliptical mirrored surface.

Classically, Fermat's principle can be considered as a mathematical consequence of Huygens' principle. Indeed, of all secondary waves (along all possible paths) the waves with the extrema (stationary) paths contribute most due to constructive interference. Supposing that light waves propagate from A to B by all possible routes ABj , unrestricted initially by rules of geometrical or physical optics. The various optical paths ABj will vary by amounts greatly in excess of one wavelength, and so the waves arriving at B will have a large range of phases and will tend to interfere destructively. But if there is a shortest route AB0, and the optical path varies smoothly through it, then a considerable number of neighboring routes close to AB0 will have optical paths differing from AB0 by second-order amounts only and will therefore interfere constructively. Waves along and close to this shortest route will thus dominate and AB0 will be the route along which the light is seen to travel.[6]*Ratio of sound speeds is the index of refractionidea of eigenrays

*Fig. 1.6. Huygens construction for diffraction of a pulse at a reflecting half plane. Incident, reflected, and diffracted pulse positions are shown for a sequence of times. The transmitted wave is omitted to simplify the sketch (i.e., R not necessarily 0 or 1)Diffraction strongest in direction of propagation (weak to sides); separate arrival in time- later.

Scattered sound used to identify hidden bodies that cause the scattering.*Point source expands and contracts, produces continuous spherical wave. Corresponding condensations (density and pressure above ambient) and rarefactions (below ambient). Similar to pulse source.

Wavelength . Point source small compared to .

Acoustic energy spread over larger and large spheres, therefore pressure amplitude decreases.

Fig. 1.7 Radiation from a very small periodically pulsating source. (a) pressure filed at an instant of time. The dark condensations are lightened at increasing range to show the decreasing acoustic pressure. (b) graph of the range-dependent pressure at an instant of time. (c) Time-dependent pressure signal at a point in space.

*Basic definitions of wavelength and period. Temporal phase, angular frequency. spatial phase and wavenumber

*Increment time and space gives speed of advance*Fig. 1.8. Geometry for the local plane wave approximation

Many sources are points. Often very far away and want to regard pressure waves as plane but how far away and over what extend (perpendicular) to propagation direction?

If within W, then within 45 degrees of phase may or may not be adequate depends on application

*Binomial(1+q)^(1/2) = 1 + q/2 q2/8 +

Fig. 1.9. Geometry for several sources with receiver at Q. The local region for the plane wave approximation is the dashed rectangle. As a plane wave approximation in the region, the incident sound pressures have the amplitudes Pn. The sources are at y0, y1, y2, yn. The distances from the sources to the listening point Q at R (angle alpha) R, R1, R2, Rn. The acoustic pressure at Q is the sum of the pressures contributed by the several sources. *Beyond critical range, no constructive or destructive interference no peaks or troughsLet alpha = 0 for simplicity, if difference in range less than lambda/2, then cant have a pressure minimum.*Carry along complex as long as useful, then discard imaginary term (or vice versa)**Fig. 1.11 Pressure differential across a small volume. (a) Lagrangian coordinates. The pressure differential causes the mass Axyz to move to the right. (b) Eularian coordinates. Mass flow is through the small volume xyz (u is the component of velocity along the x axis; A is the ambient density). *Fig. 1.11 Pressure differential across a small volume. (a) Lagrangian coordinates. The pressure differential causes the mass Axyz to move to the right. (b) Eularian coordinates. Mass flow is through the small volume xyz (u is the component of velocity along the x axis; A is the ambient density)*Fig. 1.11 Pressure differential across a small volume. (a) Lagrangian coordinates. The pressure differential causes the mass Axyz to move to the right. (b) Eularian coordinates. Mass flow is through the small volume xyz (u is the component of velocity along the x axis; A is the ambient density)

**Fig 1.12. Reflection and refraction geometry for vector particle velocities (heavy arrows) at an interface between two fluids.*Fig 1.12. Reflection and refraction geometry for vector particle velocities (heavy arrows) at an interface between two fluids.

*Fig 1.12. Reflection and refraction geometry for vector particle velocities (heavy arrows) at an interface between two fluids.

*Fig 1.12. Reflection and refraction geometry for vector particle velocities (heavy arrows) at an interface between two fluids.

*Fig 1.12. Reflection and refraction geometry for vector particle velocities (heavy arrows) at an interface between two fluids.

*Fig 1.12. Reflection and refraction geometry for vector particle velocities (heavy arrows) at an interface between two fluids.

*Fig 1.12. Reflection and refraction geometry for vector particle velocities (heavy arrows) at an interface between two fluids.

*Fig 1.12. Reflection and refraction geometry for vector particle velocities (heavy arrows) at an interface between two fluids.

*Fig 1.12. Reflection and refraction geometry for vector particle velocities (heavy arrows) at an interface between two fluids.

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Fig. 1.13. Water and sediment structure south of Long Island. Fig. 1.14. Relative amplitude R12 and phase shift of the reflection at the 1-2 interface show in in Fig 1.13*Fig. 1.15. Reflection beyond critical angle. (a) actual reflection and phase shift. (b) equivalent reflection at a virtual reflector at depth z.*Fig. 1.16. Geometry for head wave propagation at bottom sediment, The impulse source is in a lower speed medium (e.g., water over most sediments). The Huygens ssources move along the interface at the speed in the bottom (higher speed) medium.*Fig. 1.17. Impulse response fro a shallow point source of sound under a model of the Arctic ice canopy. The source was driven by 2 cycles of 62.5 kHz; =2.37 cm in water. Source and receiver depths were 0.4 in water; source and receiver weer separated by 27 ; path length in plate L = 26 cm. From Brown and Medwin et al., 1988.

Can have direct and reflected arrivals too shown in impulse response*Fig. 1.18. Reflection from a finite circular plane area. (a) circular area. (b) Phase zones. (c) Rectangular area*Ignore diffraction at disk edgeFig. 1.18. Reflection from a finite circular plane area. (a) circular area. (b) Phase zones. (c) Rectangular area

*Ignore diffraction at disck edge*Note z down herFig. 1.19. Three coordinate systems. The z axis is drawn as positive downward. The (longitude) angles are measured in planes perpendicular to the z axis. The (latitude) angle is measured to the z axis. (a) Rectangular. (b) Cylindrical. (c) Spherical.*Note z down here*Note z down here*Note z down here*Pick waves propagating in +directions*Note z down here. Re-arrrange first equation (spherical) to get second form. Then clear p ~ RP*Note z down here*Note z down here*Note z down here*Snells law, diffraction, interference, wave equation, *