Solutions to the Acoustic Wave Acoustic Wave Equation

Solutions to the
Solutions to the
Acoustic Wave
Acoustic Wave
Equation
Equation

Outline
Outline
1.
1.
Plane Wave: Dispersion relationship, freq.,
Plane Wave: Dispersion relationship, freq.,
wavelength, wavenumber, slowness, apparent
wavelength, wavenumber, slowness, apparent
velocity, apparent wavelength.
velocity, apparent wavelength.
2. Spherical Wave.
2. Spherical Wave.
3. Green’s function, asymptotic Green’s function.
3. Green’s function, asymptotic Green’s function.

Harmonic Motion:
Harmonic Motion:
P = Acos(wt)
P = Acos(wt)
Phase
Phase
Time (s)
Time (s)

T = sec/cycle
T = sec/cycle
1/T=f = cycle/s
1/T=f = cycle/s
2pi/T=w=radians/s
2pi/T=w=radians/s
w=2pi/T
w=2pi/T
Period
Period

Harmonic Ripples:
Harmonic Ripples:
P = Acos(kx)
P = Acos(kx)
Phase
Phase
Time (s)
Time (s)
k=2pi/
k=2pi/
x
x
Wavenumber
Wavenumber
Wavelength
Wavelength

Plane Wave Solution
Plane Wave Solution

Plug (2) into (1)
Plug (2) into (1)
P
P
P =
P =

c
c
2
2
2
2
;
κ
ρ
ρ
c =
c =
2
2
(1)
(1)
P = Ae
P = Ae
i(kx-wt)
i(kx-wt)
(2)
(2)
(k - w ) P = 0

(k - w ) P = 0
2
2
2
2
c
c
2
2
dO
dO
dt
dt
= kdx/dt –w =0
= kdx/dt –w =0
dx/dt =w/k
dx/dt =w/k
wavenumber
k = w
k = w
c
c
angular frequency
k = w
c
= 2 pi
= 2 pi
Phase=O
Phase=O
Wavefront = Line of constant phase

Wavelength=shortest distance between adjacent peaks
Faster Velocities = Stiffer Rocks

2D Plane Wave Solution
2D Plane Wave Solution
P = Ae
P = Ae
i(k x + k z - wt)
i(k x + k z - wt)
(2)


x
x
z
z
=
Ae
Ae
i(k r - wt)
i(k r - wt)
(2)


Equation of a line: k r = cnst
Equation of a line: k r = cnst
k
Phase
Phase
= |k|(sin , cos )

= |k|(sin , cos )
k = (k , k )
k = (k , k )
x
x
z
z

k
r
(x,z)
x
z
r
k
O
= |r||k| cos(O)
= constant
Any pt along line phase is cnst
k is Perpendicular to Wavefront
k is Perpendicular to Wavefront

2D Plane Wave Solution
2D Plane Wave Solution
P = Ae
P = Ae
i(k x + k z - wt)
i(k x + k z - wt)
(2)

x
x
z
z
=
Ae
Ae
i(k r - wt)
i(k r - wt)
(2)


Equation of a line: k r = cnst
Equation of a line: k r = cnst
k
Phase
Phase
= |k|(sin , cos )
= |k|(sin , cos )
= 2pi/|k|
= 2pi/|k|
k = (k , k )
k = (k , k )
x
x
z
z
x
x

=
=
sinO
sinO
z
z
z
z
=
=
cosO
cosO

Time (s)
Time (s)
Apparent Velocity
Apparent Velocity
z
z
x
x
dz/dt=apparent V
dz/dt=apparent V
z
z
=
=
z
z
T

T
dx/dt=apparent V
dx/dt=apparent V
x
x
=
=
x
x
T
T
sinO
sinO
=
=
T
T

Time (s)
Time (s)
Examples: dx/dt = v/sinO
Examples: dx/dt = v/sinO
Examples: dx/dt = v/sinO=v
Examples: dx/dt = v/sinO=v
O=90
=
=
z
z
x

x
=
=

Examples: dx/dt = v/sinO
Examples: dx/dt = v/sinO
=
=
z
z
=
=
O=0
x
x
=
=
Time (s)
Time (s)

Time (s)
Time (s)
0.0
0.0
0.45
0.45
0.0
0.0
250
250

X (ft)
X (ft)

Outline
Outline
1.
1.
Plane Wave: Dispersion relationship, freq.,
Plane Wave: Dispersion relationship, freq.,
wavelength, wavenumber, slowness, apparent
wavelength, wavenumber, slowness, apparent
velocity, apparent wavelength.
velocity, apparent wavelength.
2. Spherical Wave.
2. Spherical Wave.
3. Green’s function, asymptotic Green’s function.
3. Green’s function, asymptotic Green’s function.

Energy of an Acoustic Wave
Energy of an Acoustic Wave
Work Performed:
Work Performed:
du
du
dx
dx
dz
dz

dW = (Pdzdy)du = PdV
dW = (Pdzdy)du = PdV
dy
dy


W = (Pdzdy)du = PdV
W = (Pdzdy)du = PdV


but dV = VdP/
but dV = VdP/
κ
κ


W = PdP
W = PdP


κ
κ
V
V
= P
= P


κ
κ

V
V
2
2
2
2
Hooke’s Law dP = dV/V
Hooke’s Law dP = dV/V
κ
κ
F=Pdzdy
F=Pdzdy

r
Spherical Wave in
Spherical Wave in
Homogeneous Medium
Homogeneous Medium
1
2
3
4
Ae
Ae
i(k r - wt)
i(k r - wt)
(2)


r

P =
P =
P
P
P =
P =

c
c
2
2
2
2
satisfies
satisfies
except at origin
except at origin
Geometrical speading
Geometrical speading
Ray is traced such that it is always
Ray is traced such that it is always
Perpendicular to wavefront
Perpendicular to wavefront
r = x + y + z
r = x + y + z
2
2
2
2
2

2
r is distance between pt source
and observer at (x,y,z)

Outline
Outline
1.
1.
Plane Wave: Dispersion relationship, freq.,
Plane Wave: Dispersion relationship, freq.,
wavelength, wavenumber, slowness, apparent
wavelength, wavenumber, slowness, apparent
velocity, apparent wavelength.
velocity, apparent wavelength.
2. Spherical Wave.
2. Spherical Wave.
3. Green’s function, asymptotic Green’s function.
3. Green’s function, asymptotic Green’s function.

r
1
2
3
4
Ae
Ae
i(w
i(w
τ
τ

- wt)
- wt)
(2)


P =
P =
P
P
P =
P =

c
c
2
2
2
2
satisfies
satisfies
except at origin
except at origin
Geometrical speading
Geometrical speading
w
w
τ
τ

kr = (kc)r/c =
kr = (kc)r/c =
Time taken along ray
w
Spherical Wave in
Spherical Wave in
Heterogeneous Medium
Heterogeneous Medium
Valid at high w and smooth media
Valid at high w and smooth media
i i i i
i
i

Summary
Summary
2. k=|k|(sin O, cos O)
2. k=|k|(sin O, cos O)


k
k
z
z


cos O
cos O
x
x

sin O
sin O
;
;
z
z
x
x
V =
V =
z
z
V =
V =
T
T
T
T
x
x
;
;
P = Ae
P = Ae
i(kx-wt)
i(kx-wt)
1.
1.

V =
V =
T
T
κ
ρ
ρ


=
=
=
=

Summary
Summary
3. k=|k|(sin O, cos O)
3. k=|k|(sin O, cos O)


p
p
w
w
w
w
= p
= p
Slowness Vector
Slowness Vector

4. Motivation: Spatial aliasing
4. Motivation: Spatial aliasing
X <
X <
x
x
2
2
Geophone sampling interval
Geophone sampling interval