Homework Assignment
assigned:
due:
10 Jan 2013
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/
1
Oc679 Acoustical Oceanography
Next few lectures:
Lighthill Ch.1
Medwin&Clay Ch.2
Wave physics
acoustic intensity / pressure
spreading / reflection / refraction
( propagation properties )
interference
( “near-field” effects )
definition of the “far-field”
wave properties
the wave equation
acoustic impedance
reflection / transmission … quantified
some properties
head waves
Doppler sonar
Sonar Equation
define unit of measurement for acoustics
absorption
transmission losses
example
tomography
2
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
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Oc679 Acoustical Oceanography
4
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 MHz
15 m
5
p, acoustic pressure
(unit [Pa])
pA = ρgh +
atmospheric
pressure +
waves +
nonhydrostatic
effects
time
sound pressure
silence
pA, total ambient
pressure
(unit [Pa])
sound
1 m water ≈ 104 Pa
total pressure
atmospheric pressure ≈ 105 Pa
large amplitude internal wave ≈ 100 Pa
0
fin whale (100 m range) ≈ 10 Pa
ATOC source level (75 Hz) ≈ 104 Pa (200 dB)
spherical spreading
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
4πR2·iR = 4πR02·i0
iR, io are acoustic intensities at R, Ro
pulse has duration t
Therefore,
io Ro 2
iR  2
R
Sound intensity decreases as 1/R2 – this is termed spherical spreading
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Oc679 Acoustical Oceanography
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 – phasedependent
4. Diffraction – when sound encounters an obstacle some
of the energy bends around it, some is reflected
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Oc679 Acoustical Oceanography
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 timet, wavelets originating at wave front a, travel R to
b, which is now the location of the new wave front
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Oc679 Acoustical Oceanography
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):
sin 1
c1

sin  2
c2
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
Huygens’ principle demonstrates
laws of reflection and refraction
reflection/refraction applet
http://webphysics.ph.msstate.edu/jc/library/24-2/huygens.htm
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Oc679 Acoustical Oceanography
wave refraction
which is greater c1? or c2?
11
wave refraction
c1 > c2
you can tell this by considering that
frequency is invariant
f = c1/λ1 = c2/λ2 = constant
rays bend towards lower c medium
12
applet example – waves had traveled from a distant point such
that wave curvature was negligible (plane wave approximation)
below is a spherical wave
reflected wave fronts
point source
above a half-plane
reflected wave fronts
appear to come from
an image source in
lower half-plane
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Oc679 Acoustical Oceanography
wave front interpretation
14
ray interpretation
15
sound speed
profile
refracted / refracted
refracted / surface-reflected
surface-reflected / bottom-reflected
these are computed from ray theory –
integration of odes
initial condition is the ray take-off angle
this figure shows the paths from many take-off angles
16
sound propagation paths in the ocean
c(z)
c(z)
17
sound propagation paths in the ocean
nearly isothermal
stratif
c(z)
c(z)
18
sound propagation paths in the ocean
c(z)
c(z)
19
single slit diffraction
20
single slit diffraction
radially-spreading wave
from a plane wave
21
Diffraction – obstacle effects
obstacle
http://www.phy.hk/wiki/englishhtm/Diffraction2.htm
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Oc679 Acoustical Oceanography
Interference effects
Phase
fT [cycles], 2fT [radians]  temporal phase
2R/ [radians]
 spatial phase
interference effects from
local source / source array
may be important in the
near-field of the source
23
Interference effects
interference effects from
local source / source array
may be important in the
near-field of the source
192 element array Urick Fig 3.3
near-field of an ultrasonic transducer
http://www.ndt-ed.org/EducationResources/CommunityCollege/Ultrasonics/EquipmentTrans/radiatedfields.htm
24
Interference effects
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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Oc679 Acoustical Oceanography
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
pT  p A  p, p  p A
T   A   ,    A
subscript A refers to the ambient pressure, density,
which are constant
p,  are acoustic pressure, density
Newton’s 2nd Law for Acoustics  F = ma
pressure across a fluid CV
thru which acoustic wave
travels 
applies at point x and time t

p
u
 A
x
t
 rate at which CV is accelerated
[w1]
so p, du/dt in quadrature
p,u out of phase by π
Conservation of Mass for Acoustics
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
 A
u 

x t
[w2]
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Oc679 Acoustical Oceanography
Equation of State for Acoustics
Hooke’s law for an elastic body  stress  strain
force per
unit area
relative change
in dimension
For acoustics
• stress is the acoustic pressure, p
• strain is the relative change of density, /A
• proportionality constant is bulk modulus of elasticity, E
p(
E
A
)
[w3]
this is equivalent to an acoustical equation of state
1D wave equation
Eliminating u in [w1, w2] and using w3, we get the 1D linear acoustic wave equation
2 p  A 2 p

x 2
E t 2
[w4]
alternatively, we could have eliminated p rather than  using [w3] and obtained an
equation for the acoustic density
This can be derived for the particle velocity, u, or particle displacement or other
parameters characteristic of the wave - p used as hydrophones are pressuresensitive
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Oc679 Acoustical Oceanography
wave equation has solutions of form p  Pe
substituting plane wave solution
into wave equation gives 
c2 
E
A
i
recall e  cos  i sin
i ( t  kx )
(remember c 

k
)
2
2
so we can write the wave equation as  p  1  p
x 2 c 2 t 2
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
[w5]
this is shown by substitution
u
 iU
t
u
 ikU
x
u
 u
u

 c
t
k x
x
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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Oc679 Acoustical Oceanography
acoustic Mach number
from [w2] and
 A
u 

x t
u
u
 c
can determine ratio of acoustic particle velocity to sound speed
t
x
M
u 

c A
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
acoustic pressure-density relation
combining p  ( Ac )u and M 
u 

c A

p  c 2
c=√p/ρ
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/
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Oc679 Acoustical Oceanography
acoustic intensity
defined as the energy per unit time [ power ] - passing through a unit area
a wave traveling in the +x direction has intensity defined by the product of the
instantaneous values of acoustic pressure and the particle velocity
- since c is not a function of
i x  p(t  x / c )u x (t  x / c )
direction, no subscript x needed
( i x  pu x )
using the equation for acoustic impedance
note: units are W/m2
p2
[ p(t  x / c )]2
ix 
ix 
 Ac
 Ac
with the long range plane wave approximation (replace x with R)
[ p(t  R / c )]2
ix 
 Ac
for a sinusoidal wave p  P sin(kx  t )
P2
1  cos[2(kx  t )]
ix 
sin2 (kx  t )  P 2
 Ac
2 Ac
instantaneous intensity ix oscillates
between 0 and P2/(Ac) at frequency 2
average intensity - time average at x
Prms 2
P2
this is alternatively  U2 via
Ix  i x 

acoustic impedance equation
2 Ac  Ac
where P is peak pressure, P =2Prms
note: analogy to electronics in
which Power = V2/Z
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Oc679 Acoustical Oceanography
summary – sound wave physics
p
u
 A
x
t
u 
 A

x t
E
p  ( )
A
conservation of momentum 
conservation of mass
acoustical equation of state
1-D wave equation
2 p  A 2 p

2
x
E t 2
solutions
p  Pei ( t kx )
u  Ue i (t kx )
property of the medium
rho-c is acoustic impedance
relationship of c to properties of the medium
average acoustic intensity
property of the wave
p  (  Ac )u
p  c 2
Ix  i x
or
c=√p/ρ
Prms 2
P2


2 Ac  Ac
31
Reflection and Transmission at interfaces
plane waves
boundary conditions
1. pressures equal on each side of interface
2. normal components of particle velocity equal
on each side of interface
pi  pr  pt
uzi  uzr  uzt
ui
ur
ut
normal components of particle velocity are
uzi  ui cos1
uzr  ur cos1
uzt  ut cos 2
replacing u with p using acoustic impedance relationship
p
uzi  i cos 1
1c1
uzr  
uzt 
pr
cos 1
1c1
pt
cos  2
 2c 2
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Oc679 Acoustical Oceanography
define reflection and transmission coefficients
p
p
R12  r
T12  t
pi
pi
pressure boundary condition 
1  R12  T12
velocity boundary condition 
2c2 (1  R12 )cos1  1c1T12 cos 2
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
 c cos1  1c1 cos2
22c2 cos1
R12  2 2
T12 
2c2 cos1  1c1 cos2
2c2 cos1  1c1 cos2
example  source beneath water-air interface which is unrealistically smooth is
normally incident to the interface (cos1=1, cos2=1)
air 1 kg/m3, cair  330 m/s, water1000 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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Oc679 Acoustical Oceanography
we can also get total reflection for sufficiently large incident angles into higher c medium
c
Snell’s law gives 2  arcsin( 2 sin1 )
c1
1/ 2
2
 c 

use sin2+cos2 =1
2
2
Snell’s law can be written cos 2  1    sin 1 
2
  c1 

 c2 
2
when 1    sin 1  0 cos2 is complex
 c1 
c
this occurs when 1  arcsin 1  c where subscript c refers to a critical angle
c2
when 1 < c, R12 is real and |R12| < 1
(lossy medium)
when 1 > c, R12 is imaginary and |R12| = 1
1c1g 2
but with a phase shift given by


arctan
2 i 
2c2 cos1
and R12  e
1=1033 kg/m3
c1=1508 m/s
 c 

g 2   2  sin2 1  1
 c1 

2
1/ 2
R12
2=2 1
c2=1.12c1

if one is interested in getting acoustic signal across the interface
(i.e., across the bottom sediments), R12 is a bottom loss
34
Oc679 Acoustical Oceanography
Reflection and Transmission at multiple thin layers
where layer thicknesses small compared to distances to source/receiver, can
use plane wave assumption
the 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=kwavehicosI, where hi is layer thickness
M&C section 2.6.3
an 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)
0m
using c = 1500m/s we can convert
time scale to range or depth scale
(at least to the seafloor, below
which c >> 1500)
water
sediments
37.5 m
2nd reflection
bedrock
range = ct/2
t is the total travel time from
source back to receiver
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Oc679 Acoustical Oceanography
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 sinc=c1/c2
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
properties can be deduced by Huygens
wavelets construction
here c2=2c1, c=30
at angles > c sound pressures and
displacements are equal on both sides of
interface and are excited at points hi
moving as the refracted wave along the
interface at c2
the head wave front is the envelope of
the wavelets
for a spherical point source, head wave
is represented by a conical surface in 3D
36
Oc679 Acoustical Oceanography
what does an arrival look like?
axial arrivals
short path at low speed
less attenuation due to
cylindrical spreading
no reflective losses
interfering reflected arrivals
long path length at high speed
attenuated by reflection at top/bottom
head wave
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Oc679 Acoustical Oceanography
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)
T has pulled ahead of D, R
head wave is 1st arrival near interface, but further away,
direct wave arrives 1st
38
note scale change
source: Computational Ocean Acoustics
Oc679 Acoustical Oceanography
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/s
c2=8000 m/s
39
Oc679 Acoustical Oceanography
change of phase upon
reflection at free surface
source
40
Oc679 Acoustical Oceanography
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Oc679 Acoustical Oceanography
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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Oc679 Acoustical Oceanography
43
Oc679 Acoustical Oceanography
44
Oc679 Acoustical Oceanography
what does an arrival look like?
axial arrivals
short path at low speed
less attenuation due to
cylindrical spreading
no reflective losses
interfering reflected arrivals
long path length at high speed
attenuated by reflection at top/bottom
head wave
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Oc679 Acoustical Oceanography
summary – sound wave physics
p
u
 A
x
t
u 
 A

x t
E
p  ( )
A
conservation of momentum 
conservation of mass
acoustical equation of state
1-D wave equation
2 p  A 2 p

2
x
E t 2
solutions
p  Pei ( t kx )
u  Ue i (t kx )
property of the medium
rho-c is acoustic impedance
relationship of c to properties of the medium
average acoustic intensity
property of the wave
p  (  Ac )u
p  c 2
Ix  i x
or
c=√p/ρ
Prms 2
P2


2 Ac  Ac
46
Doppler sonar
such as an ADCP (acoustic Doppler current profiler)
measure of the velocity of water from the shift in frequency of a transmitted pulse
wavelength of the traveling wave is constant
λDoppler = λsource
so
fDoppler= - fsource(V/c)
fDoppler = change in returned frequency
fsource = frequency of transmitted signal
V = velocity of scatterers
c = sound speed
transducer
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
......
......
fsource
V
fsource+fDoppler
......
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Oc679 Acoustical Oceanography
acoustic travel time measurement
what can we learn from a source/receiver pair?
separated by range R
in medium with sound
speed c
t1 = R/c
source/
receiver
receiver/
reflector
t2 = R/c
if U = 0, sound speed c is determined
by measuring t1 or t2
t1 = R/c, t2 = R/c, t1 + t2 = 2R/c
t1 = R/c + R/U
source/
receiver
U
receiver/
reflector
t2 = R/c –R/U
if U  0
t1 + t2 = 2R/c
t1 - t2 = 2R/U
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Oc679 Acoustical Oceanography
49
Continuous wave sinusoidal signals
Wavelength,  is the distance
between adjacent condensations (or
adjacent rarefactions) along direction
of travel of wavefront
p( R) 
Po Ro
2 R
sin(
)
R

I just wrote this down here. It is not obvious but
it is a consequence of conservation of energy
Radiation from a
small sinusoidal
source
At a point, time between adjacent condensations, T =  /c = 1/f
or c = f 
here T [s], f [Hz],  [m], c [m/s]
Dimensionless products fT [cycles], 2fT [radians]
compare 2R/
 temporal phase
 spatial phase
Radian frequency,  = 2f = 2/T [rad/s]
Wavenumber, k = 2/ [rad/m]
c = /k
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Oc679 Acoustical Oceanography
Phase
fT [cycles], 2fT [radians]  temporal phase
2R/ [radians]
 spatial phase
51
To describe a wave propagating in the
positive R direction – outward
p ( R, t ) 
Po Ro
sin(t  kR)
R
p ( R, t ) 
Po Ro
sin[ (t  R / c)]
R
p ( R, t ) 
Po Ro
sin[2 (t / T  R /  )]
R
For a wave propagating in the negative
R direction, replace (t-kR) with (t+kR)
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Oc679 Acoustical Oceanography
Interference
 effects of phase
Multiple sources
Linear waves add algebraically
instantaneous pressure may be < or > than any individual source pressure
Local plane wave approximation
W is the local alongwavefront extent of plane
wave approximation
At large distance from source, spherical
wave appears as a plane wave (curvature is
minimal)
W2
2
2
R  (R   ) 
sagitta of the arc
4
If we require  to be  /8 for a plane
W2
2
 (2 R   )
wave, then
4
W2 
in the case R


8R 8
W2
2 R
or the region over which we can
4
consider the spherical wave to be
plane is
W  ( R)1/ 2
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Oc679 Acoustical Oceanography
plane wave limit
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Fresnel and Fraunhofer approximations - range dependencies
to 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 is
p   Pn sin(t  kRn )
n
or using trig identity
p  sin(t ) Pn cos(kRn )  cos t  Pn sin(kRn )
using
R2  y 2  x2
n
n
Rn 2  ( y  yn )2  x 2
Fraunhofer – very long range
y
Rn  R[1  n sin  ]
R
Rn 2  ( R sin   yn )2  R 2 cos 2 
so that
2 yn
yn 2 1/ 2
Rn  R(1 
sin   2 )
R
R
binomial expansion
yn
yn 2
Rn  R[1  sin   2 (1  sin 2  ) 
R
2R
Fresnel – nearer range
yn
yn 2
Rn  R[1  sin   2 (1  sin 2  )]
R
2R
]
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Oc679 Acoustical Oceanography
previous axes rotated to remove 
Far-field approximation
consider y0, y1, y2 to be different elements on the
surface of a radiator
In 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 wavelengths
Closest 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:
R2  ( R  W )
2
2 1/ 2
W2
 R(1  2 )
2R
Alternatively, we define the far
field by
W2
R

W2
or maybe R  4

To avoid destructive interference, R2-R  /2
(note that constructive interference at  is less restrictive)
so that
W2 
R2  R 
2R

2
Critical range, beyond which destructive interference
does not occur
W2
Rc 

binomial expansion: (a  x)  a  na
n
n
n 1
n(n  1) ( n  2) 2
x
a
x 
2!
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Oc679 Acoustical Oceanography
low-frequency source, say 100 Hz, with =15 m will be relatively large
maybe ½ width, W = 0.5 m
will be free of interference effects for
R4
W2

 6 meters
or a higher-frequency source like an ADCP at 100 kHz, =0.015 m
will have a smaller transducer, say W = 0.05 m, and
will be free of interference effects for
R4
W2

 0.6 meters
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Fresnel zones spherical wave reflection at an interface
source/receiver
source transmits signal, and is then quiet while receiver listens
signal is reflected in all directions from scattering elements on surface
the full calculation of
reflection consists of
integrating all wavelets
radiating as point sources
from small scattering
elements on the surface.
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)
  h  1/ 2
Fresnel zones (phase zones) defined by radii rn  
 n
2


1st Fresnel zone, within which there is a minimum of destructive
1/ 2
interference is defined by
 h 
r1  

 2 
1/ 2
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Oc679 Acoustical Oceanography
what does an arrival look like?
axial arrivals
short path at low speed
less attenuation due to
cylindrical spreading
no reflective losses
interfering reflected arrivals
long path length at high speed
attenuated by reflection at top/bottom
head wave
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Oc679 Acoustical Oceanography
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)
T has pulled ahead of D, R
head wave is 1st arrival near interface, but further away,
direct wave arrives 1st
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note scale change
source: Computational Ocean Acoustics
Oc679 Acoustical Oceanography
applet example – waves had traveled from a distant point such
that wave curvature was negligible (plane wave approximation)
below is a spherical wave
reflected wave fronts
point source
above a half-plane
reflected wave fronts
appear to come from
an image source in
lower half-plane
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Oc679 Acoustical Oceanography