1 oc679 acoustical oceanography homework assignment assigned:10 jan 2013 due:22 jan 2013 1)get...

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Oc679 Acoustical Oceanography

Homework Assignment

assigned: 10 Jan 2013due: 22 Jan 2013

1) Get seasonal ocean profiles of T,S,p from Levitus data along 150W from 45S to 45N. I think these are 1 degree resolution but 5 degrees would be fine (though not much less work). You can get these at http://ingrid.ldgo.columbia.edu/SOURCES/.LEVITUS94/. Use Seawater routines to compute and contour plot potential temperature , potential density , and sound speed, c. Focus on the upper 1500m. Comment on temporal variability.

get Oceans toolbox at http://woodshole.er.usgs.gov/operations/sea-mat/

http://ingrid.ldgo.columbia.edu/SOURCES/.LEVITUS94/

http://woodshole.er.usgs.gov/operations/sea-mat/

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Next few lectures: Lighthill Ch.1Medwin&Clay Ch.2

Wave physicsacoustic intensity / pressurespreading / reflection / refraction ( propagation properties )interference ( “near-field” effects )definition of the “far-field”wave propertiesthe wave equationacoustic impedancereflection / transmission … quantifiedsome properties

head wavesDoppler sonar

Sonar Equationdefine unit of measurement for acousticsabsorptiontransmission lossesexampletomography

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Physics of Sound Propagation – M&C Ch2

sound is a mechanical disturbance travels as longitudinal or compressive wave (in geophysics, P-wave)(compared to a transverse wave – like surface gravity waves)transverse/longitudinal wave applet

identified as an incremental acoustic pressure << ambient pressure

In a homogenous, isotropic medium, an explosion will create adjacent region of higher density, pressure – condensation pulse

By contrast, rarefaction pulse created by implosion

This pulse will move away as a spherical wave shell so that the initial energy is spread over shells of larger radii, but lower intensity - spherical spreading

absorption and scattering affect intensity

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=14

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acoustic wavelengths

f =c/f

10 Hz 150 m

100 Hz 15 m

1 kHz 1.5 m

10 kHz 15 cm

100 kHz 1.5 cm

1 MHz 1.5 mm

10 MHz 150 m

100 MHz15 m

pA, total ambient pressure (unit [Pa])

p, acoustic pressure (unit [Pa])

0

pA = ρgh + atmospheric pressure + waves + nonhydrostatic effects

silence sound

total pressure

sound pressure

1 m water ≈ 104 Pa

atmospheric pressure ≈ 105 Pa

large amplitude internal wave ≈ 100 Pa

fin whale (100 m range) ≈ 10 Pa

ATOC source level (75 Hz) ≈ 104 Pa (200 dB)

time

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Acoustic intensity = energy per unit time passing thru a unit surface area [J s-1 m-2]

Total energy is integrated over spherical surface 4πR2

Conservation of energy

iR, io are acoustic intensities at R, Ro

pulse has duration t

2

2o o

R

i Ri

RTherefore,

Sound intensity decreases as 1/R2 – this is termed spherical spreading

spherical spreading

4πR2·iR = 4πR02·i0

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Modifications to wave propagation

1. Reflection – wave incident on boundary

2. Refraction – change in sound speed changes direction of wave propagation as well

3. Interference – combination of sound waves – phase-dependent

4. Diffraction – when sound encounters an obstacle some of the energy bends around it, some is reflected

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Huygens’ principle –useful for geometrical construction of reflection, refraction and diffraction

Consider each point on an advancing front as a source of secondary waves, each moving outward as spherical wavelets – the outer surface that envelops these waves represents the new wave front

R c t

In timet, wavelets originating at wave front a, travel R to b, which is now the location of the new wave front

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http://webphysics.ph.msstate.edu/jc/library/24-2/huygens.htm

Huygens’ principle demonstrates laws of reflection and refractionreflection/refraction applet

Law of reflection: angle of reflection of rays ( wave fronts)= angle of incidence, and is in the same plane

Law of refraction (Snell’s Law): 1 2

1 2

sin sin

c c

where 1, 2 are angles measured between rays and normal to interface or between wave fronts and interface

c1, c2 are the sounds speeds in the 2 media

M&C (fig 2.2.3) show a sketch of the case c2>c1

http://www.walter-fendt.de/ph14e/huygenspr.htm

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wave refraction

which is greater c1? or c2?

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wave refraction

c1 > c2you can tell this by considering that frequency is invariant f = c1/λ1 = c2/λ2 = constant

rays bend towards lower c medium

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applet example – waves had traveled from a distant point such that wave curvature was negligible (plane wave approximation)

below is a spherical wave

point source above a half-plane

reflected wave fronts

reflected wave fronts appear to come from an image source in lower half-plane

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wave front interpretation

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ray interpretation

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sound speed profile refracted / refracted

refracted / surface-reflectedsurface-reflected / bottom-reflected

these are computed from ray theory – integration of odesinitial condition is the ray take-off anglethis figure shows the paths from many take-off angles

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c(z) c(z)

sound propagation paths in the ocean

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c(z) c(z)

nearly isothermal

stratified

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c(z) c(z)


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single slit diffraction

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single slit diffraction

radially-spreading wave from a plane wave

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Diffraction – obstacle effects

http://www.phy.hk/wiki/englishhtm/Diffraction2.htmobstacle

http://www.phy.hk/wiki/englishhtm/Diffraction2.htm

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Interference effects

Phase

fT [cycles], 2fT [radians] temporal phase 2R/ [radians] spatial phase

interference effects from local source / source array may be important in the near-field of the source

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http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/EquipmentTrans/radiatedfields.htm

near-field of an ultrasonic transducer


192 element array Urick Fig 3.3

interference effects from local source / source array may be important in the near-field of the source

http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/EquipmentTrans/radiatedfields.htm

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Lloyd’s Mirror Effect (optics) has an analogue in underwater acoustics(surface interference effect) this is a straightforward case of interference of acoustic signals in which one of the sources is the surface reflection of the source wave

the result is an interference pattern with peaks and troughs in signal intensity along range R. Ultimately, I decreases as 1/R2.

2.4.2 – 2.4.4 (M&C)


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Sound Wave Physics

we consider a small region far from an oscillating spherical source where plane wave approximation holds - direction of propagation is x or R

,

,T A A

T A A

p p p p p

subscript A refers to the ambient pressure, density, which are constantp, are acoustic pressure, density

Conservation of Mass for Acoustics

A

u

x t

Here the compressibility of the fluid, however small, is important – more mass can flow into a CV than out, resulting in a net density change in the CV

Newton’s 2nd Law for Acoustics F = ma

A

p u

x t

pressure across a fluid CVthru which acoustic wave travels

applies at point x and time t

rate at which CV is accelerated [w1]

[w2]

so p, du/dt in quadraturep,u out of phase by π

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Equation of State for Acoustics

Hooke’s law for an elastic body stress strain

For acoustics• stress is the acoustic pressure, p• strain is the relative change of density, /A

• proportionality constant is bulk modulus of elasticity, E

this is equivalent to an acoustical equation of state

( )A

Ep

1D wave equationEliminating u in [w1, w2] and using w3, we get the 1D linear acoustic wave equation

[w3]

2 2

2 2Ap p

x E t

[w4]

alternatively, we could have eliminated p rather than using [w3] and obtained an equation for the acoustic densityThis can be derived for the particle velocity, u, or particle displacement or other parameters characteristic of the wave - p used as hydrophones are pressure-sensitive

force per unit area

relative change in dimension

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wave equation has solutions of form ( )i t kxp Pe cos sinie irecall

substituting plane wave solution into wave equation gives

2

A

Ec

2 2

2 2 2

1p p

x c tso we can write the wave equation as [w5]

acoustic impedance

plane waves of form ( )i t kxu Ue satisfy

u u

ct x

this is shown by substitution

ui U

tu

ikUxu u u

ct k x x

substitution into [w1]

A

p uc

x 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

(remember )

ck

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u u

ct x

acoustic Mach number

from [w2] and can determine ratio of acoustic particle velocity to sound speed

A

uM

c

where M is a kind-of Mach number, a measure of the strength of the sound wave and thereby the linearity of the signal propagation – interesting and important effects for high M

( )Ap c u

A

uM

c

acoustic pressure-density relation

combining and 2p c

this means that c can be computed from an equation of state for seawater

c=c(S,T,p)

[this is included in seawater routines]http://sea-mat.whoi.edu/

A

u

x t

c=√p/ρ

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acoustic intensity

defined as the energy per unit time [ power ] - passing through a unit areaa wave traveling in the +x direction has intensity defined by the product of the instantaneous values of acoustic pressure and the particle velocity

2[ ( / )]

x

A

p t x ci

c

using the equation for acoustic impedance

- since c is not a function of direction, no subscript x needed

with the long range plane wave approximation (replace x with R)

2[ ( / )]

x

A

p t R ci

c

note: units are W/m2

for a sinusoidal wave sin( )p P kx t

2

2 2 1 cos[2( )]sin ( )

2x

A A

P kx ti kx t P

c cinstantaneous intensity ix oscillates between 0 and P2/(Ac) at frequency 2

average intensity - time average at x 22

2rms

x x

A A

PPI i

c c

where P is peak pressure, P =2Prms

note: analogy to electronics in which Power = V2/Z

this is alternatively U2 via acoustic impedance equation

( / ) ( / )x x

i p t x c u t x c

x xi pu( )

2

x

A

pi

c

summary – sound wave physics

A

p u

x t

conservation of momentum

conservation of mass

acoustical equation of state

A

u

x t

( )A

Ep

2 2

2 2Ap p

x E t

1-D wave equation

solutions ( )i t kxp Pe ( )i t kxu Ue

( )Ap c u



2p c c=√p/ρor

22

2rms

x x

A A

PPI i

c c

rho-c is acoustic impedance

relationship of c to properties of the medium

average acoustic intensity

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Reflection and Transmission at interfaces

plane wavesboundary conditions

1. pressures equal on each side of interface2. normal components of particle velocity equal

on each side of interface

i r t

zi zr zt

p p p

u u u

normal components of particle velocity are

1

1

2

cos

cos

cos

zi i

zr r

zt t

u u

u u

u u

replacing u with p using acoustic impedance relationship

1

1 1

1

1 1

2

2 2

cos

cos

cos

i

zi

r

zr

t

zt

pu

c

pu

c

pu

c

uiur

ut

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define reflection and transmission coefficients

12

r

i

pR

p

12

t

i

pT

p

pressure boundary condition 12 12

1 R T

velocity boundary condition 2 2 12 1 1 1 12 2(1 )cos cosc R cT

these boundary conditions give the pressure reflection and transmission coefficients in terms of the angles of incidence & refraction and density and sound speeds in the media on each side of the interface

2 2 1 1 1 2

12

2 2 1 1 1 2

cos cos

cos cos

c cR

c c

2 2 1

12

2 2 1 1 1 2

2 cos

cos cos

cT

c c

example source beneath water-air interface which is unrealistically smooth is normally incident to the interface (cos1=1, cos2=1)

air1 kg/m3, cair 330 m/s, water1000 kg/m3, cwater 1500m/s

so that 1c1 2c2, T12 0, R12 -1 (the negative sign indicates phase – incident and reflected pressures out of phase – since wave speeds are in opposite direction)

(this is an example of total reflection due to impedance mismatch) [what happens when 1c1= 2c2?]

perfect transmission when 1c1= 2c2

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we can also get total reflection for sufficiently large incident angles into higher c medium

Snell’s law can be written

1/ 22

22

2 1

1

cos 1 sinc

c

use sin2+cos2 =1

when

1

1

2

arcsinc

c

c

2

22

1

1

1 sin 0c

ccos2 is complex

this occurs when where subscript c refers to a critical angle

when 1 < c, R12 is real and |R12| < 1 (lossy medium)

when 1 > c, R12 is imaginary and |R12| = 1but with a phase shift given by

1 1 2

2 2 1

arctancos

c g

c

1/ 22

22

2 1

1

sin 1c

gc 2

12

iR eand

R12

if one is interested in getting acoustic signal across the interface

(i.e., across the bottom sediments), R12 is a bottom loss

1=1033 kg/m3

c1=1508 m/s

2=2 1

c2=1.12c1

Snell’s law gives 2

2 1

1

arcsin( sin )c

c

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Reflection and Transmission at multiple thin layers

where layer thicknesses small compared to distances to source/receiver, can use plane wave assumptionthe net return signal is the sum of the reflection/transmission coefficients in the layers, with proper inclusion of the phase delay of the vertical wavenumber component through each layer i=kwavehicosI, where hi is layer thicknessM&C section 2.6.3an example of where this analysis might be used is shown here – bottom layers are identified by the reflections at different depths (source/receiver is 3 m below water surface – transmitted signal is 1 cycle of a 7 kHz sinusoid)

water

sediments

bedrock

using c = 1500m/s we can convert time scale to range or depth scale (at least to the seafloor, below which c >> 1500)

range = ct/2

t is the total travel time from source back to receiver

0 m

37.5 m

2nd reflection

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Head waves

we now consider spherical waves propagating into a higher speed medium adjacent to the source medium

• head wave moves at higher speed c2 along the interface and radiate into source medium• at sufficiently large ranges it will arrive ahead of the direct wave • rays propagate into source medium at critical angle given by sinc=c1/c2

properties can be deduced by Huygens wavelets constructionhere c2=2c1, c=30at angles > c sound pressures and

displacements are equal on both sides of interface and are excited at points h i moving as the refracted wave along the interface at c2

the head wave front is the envelope of the waveletsfor a spherical point source, head wave is represented by a conical surface in 3D

what happens at angles < c ? weak reflection at steep angles

angle = c ? this is the angle the ray path must take to make θ2=90°

angles > c ? pure reflection, no transmission

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head wave

interfering reflected arrivalslong path length at high speed attenuated by reflection at top/bottom

axial arrivals short path at low speed less attenuation due to cylindrical spreading no reflective losses

what does an arrival look like?

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Oc679 Acoustical Oceanographysource: Computational Ocean Acoustics

spherical wave (D) – pulse has yet to contact surface

pulse has met interface – reflected (R) and transmitted (T) waves apparent – notes:

• T has longer pulse length due to higher c • R shows critical angle effect of reduced amplitude for steep angles (earlier arrivals at interface)

note scale change

T has pulled ahead of D, Rhead wave is 1st arrival near interface, but further away, direct wave arrives 1st

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following is a simulation of acoustic waves traveling outward from the source, reflecting from a higher-velocity material below and from the free surface above

c1=6000 m/sc2=8000 m/s

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source

change of phase upon reflection at free surface

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The next animation shows the same model, but looking at greater distances and later times. In this case, the refracted wave in the lower medium is clear, the head wave can be seen to develop with a cross-over distance of about 120 km. The linearity of the head wave as it propagates upward is particularly well illustrated by the animation. There is a weak numerical artifact (which appears as a wave propagating up from the bottom of the image) due to not-quite absorbing boundary conditions. The amplitudes in this figure are greatly enhanced so that the head wave is visible; unfortunately, so are the numerical errors. Once again, click on the still image to view the animation

http://www.geol.binghamton.edu/~barker/animations.html

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44


45


head wave




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summary – sound wave physics

A

p u

x t

conservation of momentum

conservation of mass

acoustical equation of state

A

u

x t

( )A

Ep

2 2

2 2Ap p

x E t

1-D wave equation

solutions ( )i t kxp Pe ( )i t kxu Ue

( )Ap c u



2p c c=√p/ρor

22

2rms

x x

A A

PPI i

c c

rho-c is acoustic impedance

relationship of c to properties of the medium

average acoustic intensity

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Doppler sonar such as an ADCP (acoustic Doppler current profiler)

measure of the velocity of water from the shift in frequency of a transmitted pulsewavelength of the traveling wave is constant

λDoppler = λsource

so fDoppler= - fsource(V/c) fDoppler = change in returned frequencyfsource = frequency of transmitted signalV = velocity of scatterersc = sound speed

. .. ....

fsourcetransducer

. .. ....V

fsource+fDoppler

. .. ....

operation of a monostatic ADCP, or a transducer (transceiver) that generates a short pulse at fsource which propagates through the water. Signal is transmitted in all directions by small scatterers (smaller than the acoustic wavelength) – some fraction is reflected back along beam axis – ADCP senses signal with a modulated frequency fsource+fDoppler

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what can we learn from a source/receiver pair?

t1 = R/c

t2 = R/c

source/ receiver

receiver/ reflector

separated by range R in medium with sound speed c

t1 = R/c + R/U

t2 = R/c –R/U

source/ receiver

receiver/ reflector

if U = 0, sound speed c is determinedby measuring t1 or t2

t1 = R/c, t2 = R/c, t1 + t2 = 2R/c

U

if U 0 t1 + t2 = 2R/ct1 - t2 = 2R/U

acoustic travel time measurement

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Continuous wave sinusoidal signals

Wavelength, is the distance between adjacent condensations (or adjacent rarefactions) along direction of travel of wavefront

Radiation from a small sinusoidal source

2( ) sin( )o oP R Rp R

R

At a point, time between adjacent condensations, T = /c = 1/for c = f

here T [s], f [Hz], [m], c [m/s]

Dimensionless products fT [cycles], 2fT [radians] temporal phase compare 2R/ spatial phase

Radian frequency, = 2f = 2/T [rad/s]

Wavenumber, k = 2/ [rad/m] c = /k

I just wrote this down here. It is not obvious but it is a consequence of conservation of energy

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Phase

fT [cycles], 2fT [radians] temporal phase 2R/ [radians] spatial phase

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( , ) sin( )

( , ) sin[ ( / )]

( , ) sin[2 ( / / )]

o o

o o

o o

P Rp R t t kR

R

P Rp R t t R c

R

P Rp R t t T R

R

To describe a wave propagating in the positive R direction – outward

For a wave propagating in the negative R direction, replace (t-kR) with (t+kR)

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Interference effects of phaseMultiple sourcesLinear waves add algebraicallyinstantaneous pressure may be < or > than any individual source pressure

Local plane wave approximation

At large distance from source, spherical wave appears as a plane wave (curvature is minimal)

22 2

22

2

( )4

(2 )4

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WR R

WR

WR

in the case R

If we require to be /8 for a plane wave, then

2

8 8

W

R

or the region over which we can consider the spherical wave to be plane is 1/ 2( )W R

sagitta of the arc

W is the local along-wavefront extent of plane wave approximation

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plane wave limit

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Fresnel and Fraunhofer approximations - range dependenciesto add signals from several sinusoidal point sources, require distances to point of observation - let incident sound pressure amplitudes at Q be P1, P2, …

Total sound pressure at Q issin( )n n

n

p P t kR or using trig identity

sin( ) cos( ) cos sin( )n n n nn n

p t P kR t P kR

2 2 2

2 2 2

2 2 2 2

( )

( sin ) cos

n n

n n

R y x

R y y x

R R y R

using

21/ 2

2

2(1 sin )n n

n

y yR R

R R

so that

binomial expansion2

22

[1 sin (1 sin ) ]2

n nn

y yR R

R R

Fraunhofer – very long range

[1 sin ]nn

yR R

R

Fresnel – nearer range2

22

[1 sin (1 sin )]2

n nn

y yR R

R R

56


Far-field approximationconsider y0, y1, y2 to be different elements on the surface of a radiatorIn the near-field both constructive and destructive interference can occur between different elements, each radiating spherically, since their distance from a point of observation lying on the axis (R) can differ by many wavelengthsClosest point on the surface of the radiator to Q is along the axis (y0) at range R. The farthest point is at y2 from axis on radiator at range R2. Using geometry and binomial expansion:

previous axes rotated to remove

22 2 1/ 2

2 2( ) (1 )

2

WR R W R

R

binomial expansion: 1 ( 2) 2( 1)( )

2!n n n nn n

a x a na x a x

To avoid destructive interference, R2-R /2 (note that constructive interference at is less restrictive)so that 2

2 2 2

WR R

R

Critical range, beyond which destructive interference does not occur 2

c

WR

2

4W

R

or maybe

Alternatively, we define the far field by 2W

R

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low-frequency source, say 100 Hz, with =15 m will be relatively largemaybe ½ width, W = 0.5 m

will be free of interference effects for meters

or a higher-frequency source like an ADCP at 100 kHz, =0.015 mwill have a smaller transducer, say W = 0.05 m, and

will be free of interference effects for meters

2

4 6W

R

2

4 0.6W

R

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Fresnel zones spherical wave reflection at an interface

a fresnel zone is a region at the reflector such that the phases of all of the reflected wavelets (with respect to the reference phase for the shortest path) from this region have the same sign at the receiver.

phase difference across disk = 2kR - 2kh (k is wavenumber of signal)

Fresnel zones (phase zones) defined by radii

source/receiver source transmits signal, and is then quiet while receiver listenssignal is reflected in all directions from scattering elements on surface

1/ 2

1/ 2

2n

hr n

1st Fresnel zone, within which there is a minimum of destructive interference is defined by 1/ 2

1 2

hr

the full calculation of reflection consists of integrating all wavelets radiating as point sources from small scattering elements on the surface.

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head wave




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Oc679 Acoustical Oceanographysource: Computational Ocean Acoustics

spherical wave (D) – pulse has yet to contact surface

pulse has met interface – reflected (R) and transmitted (T) waves apparent – notes:

• T has longer pulse length due to higher c • R shows critical angle effect of reduced amplitude for steep angles (earlier arrivals at interface)

note scale change

T has pulled ahead of D, Rhead wave is 1st arrival near interface, but further away, direct wave arrives 1st

61


applet example – waves had traveled from a distant point such that wave curvature was negligible (plane wave approximation)

below is a spherical wave

point source above a half-plane

reflected wave fronts

reflected wave fronts appear to come from an image source in lower half-plane

1 oc679 acoustical oceanography homework assignment assigned:10 jan 2013 due:22 jan 2013 1)get...

Documents

plane wave slide

wave refraction c

compressive wave

transverse wave

ocean slide

rays wave fronts

field wave properties

lower c medium slide