Transcript Lec2

Solutions to the
Acoustic Wave
Equation
Outline
1. Plane Wave: Dispersion relationship, freq.,
wavelength, wavenumber, slowness, apparent
velocity, apparent wavelength.
2. Spherical Wave.
3. Green’s function, asymptotic Green’s function.
Harmonic Motion:
Phase
P = Acos(wt)
Time (s)
w=2pi/T
Period
T = sec/cycle
1/T=f = cycle/s
2pi/T=w=radians/s
Phase
Time (s)
Harmonic Ripples:
P = Acos(kx)
Wavenumber
k=2pi/
Wavelength
x
Plane Wave Solution
..
2
2
k
2
P
c
c = r ; P = Phase=O
Faster Velocities = Stiffer Rocks
i(kx-wt)
P = Ae
Plug (2) into (1)
2
(k - w ) P = 0
c2
(2)
dO =dx/dt
=w/k
kdx/dt
–w =0
dt
wavenumber
2
(1)
angular frequency
pi
k = wk ==2w
c
c
Wavefront = Line of constant phase
Wavelength=shortest distance between adjacent peaks
2D Plane Wave Solution
Phase
P = Ae
i(k x + k z - (2)
wt)
x
z
i(k r - wt)
= Ae
(2)
k = (kx , k z) = |k|(sin , cos )
x= sinO
z
k
z= cosO
Equation of a line: k r = cnst
= 2pi/|k|
Apparent Velocity
Time (s)
dx/dt=apparent V x =
x
dz/dt=apparent Vz =
z
z
T
x
T
=
T sinO
Time (s)
Examples: dx/dt = v/sinO
v/sinO=v
x=
z
=
O=90
Time (s)
Examples: dx/dt = v/sinO =
x
z
=
=
O=0
0.0
Time (s)
0.45
0.0
X (ft)
250
Outline
1. Plane Wave: Dispersion relationship, freq.,
wavelength, wavenumber, slowness, apparent
velocity, apparent wavelength.
2. Spherical Wave.
3. Green’s function, asymptotic Green’s function.
Energy of an Acoustic Wave
du
Work Performed:
W = (Pdxdy)du = PdV
Hooke’s Law dP = k dV/V
but dV = VdP/k
W =V PdP
k
2
=VP
2 k
P=
Spherical Wave in
Homogeneous Medium
i(k r - wt)
(2) ..
2
2
Ae
satisfies
r
Geometrical speading
P = c
P
except at origin
Ray is traced such that it is always
Perpendicular to wavefront
r
r = x2 + y2 + z 2
1
2
3
4
Outline
1. Plane Wave: Dispersion relationship, freq.,
wavelength, wavenumber, slowness, apparent
velocity, apparent wavelength.
2. Spherical Wave.
3. Green’s function, asymptotic Green’s function.
P=
Spherical Wave in
Heterogeneous Medium
i(wt - wt)
(2) ..
2
2
Ae
satisfies
Geometrical speading
P = c
P
except at origin
w
kr = (kc)r/c = wt
r
Time taken along ray
1
2
Valid at high w and smooth media
3
4
Summary
i(kx-wt)
1. P = Ae
V =
=
T
z
= cos O ;
Vz =
z
T
;
r
k
2. k=|k|(sin O, cos O)
-1
k
-1
x
= sin O
Vx =
x
T
Summary
3. k=|k|(sin O, cos O) = p
w
w
Slowness Vector
4. Motivation: Spatial aliasing
X<
x
2
Geophone sampling interval
p