t s = x/3.2, t p = x/5.5.
(47)
The difference in travel times, which is also the difference in
arrival times,
t s − t p = x(1/3.2 − 1/5.5) = x/7.6,
(48)
is thus a function of the distance between the source and the
receiver. Because the S wave arrives about 8 s after the P wave,
the earthquake is about 60 km away, in agreement with the distance found by an earthquake location program using arrival
times from many seismic stations. This S − P travel time technique gives an estimate of the distance from the seismometer to
the earthquake, but does not yield the azimuth and hence the
location. 1 Given S − P times at several stations, the location can
be found from the requirement that the earthquake must be a
specific distance from each station. Schematically, this method
can be thought of as locating the point on a map where arcs
of circles with the appropriate radii intersect. The problem is
actually more interesting, because the earthquake need not
have occurred at the earth’s surface.
2.4.5 Energy in a plane wave
Like waves on a string (Section 2.2.4), seismic waves transport
energy both as kinetic energy and as strain, or potential, energy.
To find this energy, consider harmonic plane S and P waves
traveling in the z direction. An SH wave with displacement in
the y direction is
u y (z, t) = B cos (ωt − kz),
(49)
where this expression is written directly in terms of displacement, rather than potential. We will see shortly that this is a
useful approach for SH waves.
The kinetic energy in a volume V is the integral of the sum
of the kinetic energy associated with each component of the
displacement
KE
u
t
dV
V
i
,
=
⎛
⎝
⎜
⎞
⎠
⎟
1
2
2
Ύ
ρ
∂
∂
(50)
because the mass is m = ρdV. Hence for the plane wave
(Eqn 49), the kinetic energy per unit wave front averaged over a
wavelength λ is
KE
B
t kz dz
B
B
sin (
)
/ .
=
−
=
=
1
2
1
2
2
4
2 2
0
2
2 2
2 2
λ
ρ ω
ω
λ
ρ ω
λ
ω ρ
λ
Ύ
(51)
The strain energy (Eqn 2.3.88) is
W
ed V
V
ij ij
.
=
1
2 Ύ
σ
(52)
Because the only nonzero strain components are
e 32 = e 23 =
1
2
∂
∂
u
z
y = Bk sin (ωt − kz)/2,
(53)
The only nonzero stress components are
σ 32 = σ 23 = µBk sin (ωt − kz),
(54)
and the strain energy per unit area of wave front averaged over
a wavelength in the propagation direction is
W =
1
2
0
λ
λ
Ύ
µB 2 k 2 sin 2 (ωt − kz)dz = µB 2 k 2 /4 = B 2 ω 2 ρ/4, (55)
where the last expression used the fact that µ = β 2 ρ and
βk = ω. Thus the strain energy and kinetic energy averaged
over a wavelength are equal, as we found for the string. Hence
the total energy averaged over a wavelength is
E = KE + W = B 2 ω 2 ρ/2,
(56)
and the average energy flux in the propagation direction is
found by multiplying by the velocity
0 = B 2 ω 2 ρβ/2.
(57)
The total energy and flux are proportional to the square of
the amplitude and the frequency, so for waves of the same
amplitude, the higher-frequency wave transports more energy.
Similarly, a plane P wave propagating in the z direction,
described by the scalar potential
φ(z, t) = A exp (i(ωt ± kz))
(58)
has a displacement which is the gradient of the potential,
u(z, t) = ∇
∇ ∇
∇ ∇φ(z, t) = (0, 0, −ik) A exp (i(ωt − kz)),
(59)
with real part
u z (z, t) = Ak sin (ωt − kz).
(60)
Using Eqn 50, the kinetic energy per unit wave front averaged
over a wavelength is
KE =
1
2
2 2 2
0
λ
ρ
ω
λ
A k
Ύ
cos 2 (ωt − kz)dz = A 2 ω 2 k 2 ρ/4.
(61)
To find the strain energy (Eqn 52), we note that the only
nonzero stress component is
σ zz = (λ + 2µ)e zz = ρα 2 e zz ,
(62)
2.4 Seismic waves 61
1 An analogous method is used to estimate that a thunderstorm is a mile away for
every 5 s between seeing lightning and hearing thunder, because light travels much
faster than sound (about 330 m/s in air).
(47)
The difference in travel times, which is also the difference in
arrival times,
t s − t p = x(1/3.2 − 1/5.5) = x/7.6,
(48)
is thus a function of the distance between the source and the
receiver. Because the S wave arrives about 8 s after the P wave,
the earthquake is about 60 km away, in agreement with the distance found by an earthquake location program using arrival
times from many seismic stations. This S − P travel time technique gives an estimate of the distance from the seismometer to
the earthquake, but does not yield the azimuth and hence the
location. 1 Given S − P times at several stations, the location can
be found from the requirement that the earthquake must be a
specific distance from each station. Schematically, this method
can be thought of as locating the point on a map where arcs
of circles with the appropriate radii intersect. The problem is
actually more interesting, because the earthquake need not
have occurred at the earth’s surface.
2.4.5 Energy in a plane wave
Like waves on a string (Section 2.2.4), seismic waves transport
energy both as kinetic energy and as strain, or potential, energy.
To find this energy, consider harmonic plane S and P waves
traveling in the z direction. An SH wave with displacement in
the y direction is
u y (z, t) = B cos (ωt − kz),
(49)
where this expression is written directly in terms of displacement, rather than potential. We will see shortly that this is a
useful approach for SH waves.
The kinetic energy in a volume V is the integral of the sum
of the kinetic energy associated with each component of the
displacement
KE
u
t
dV
V
i
,
=
⎛
⎝
⎜
⎞
⎠
⎟
1
2
2
Ύ
ρ
∂
∂
(50)
because the mass is m = ρdV. Hence for the plane wave
(Eqn 49), the kinetic energy per unit wave front averaged over a
wavelength λ is
KE
B
t kz dz
B
B
sin (
)
/ .
=
−
=
=
1
2
1
2
2
4
2 2
0
2
2 2
2 2
λ
ρ ω
ω
λ
ρ ω
λ
ω ρ
λ
Ύ
(51)
The strain energy (Eqn 2.3.88) is
W
ed V
V
ij ij
.
=
1
2 Ύ
σ
(52)
Because the only nonzero strain components are
e 32 = e 23 =
1
2
∂
∂
u
z
y = Bk sin (ωt − kz)/2,
(53)
The only nonzero stress components are
σ 32 = σ 23 = µBk sin (ωt − kz),
(54)
and the strain energy per unit area of wave front averaged over
a wavelength in the propagation direction is
W =
1
2
0
λ
λ
Ύ
µB 2 k 2 sin 2 (ωt − kz)dz = µB 2 k 2 /4 = B 2 ω 2 ρ/4, (55)
where the last expression used the fact that µ = β 2 ρ and
βk = ω. Thus the strain energy and kinetic energy averaged
over a wavelength are equal, as we found for the string. Hence
the total energy averaged over a wavelength is
E = KE + W = B 2 ω 2 ρ/2,
(56)
and the average energy flux in the propagation direction is
found by multiplying by the velocity
0 = B 2 ω 2 ρβ/2.
(57)
The total energy and flux are proportional to the square of
the amplitude and the frequency, so for waves of the same
amplitude, the higher-frequency wave transports more energy.
Similarly, a plane P wave propagating in the z direction,
described by the scalar potential
φ(z, t) = A exp (i(ωt ± kz))
(58)
has a displacement which is the gradient of the potential,
u(z, t) = ∇
∇ ∇
∇ ∇φ(z, t) = (0, 0, −ik) A exp (i(ωt − kz)),
(59)
with real part
u z (z, t) = Ak sin (ωt − kz).
(60)
Using Eqn 50, the kinetic energy per unit wave front averaged
over a wavelength is
KE =
1
2
2 2 2
0
λ
ρ
ω
λ
A k
Ύ
cos 2 (ωt − kz)dz = A 2 ω 2 k 2 ρ/4.
(61)
To find the strain energy (Eqn 52), we note that the only
nonzero stress component is
σ zz = (λ + 2µ)e zz = ρα 2 e zz ,
(62)
2.4 Seismic waves 61
1 An analogous method is used to estimate that a thunderstorm is a mile away for
every 5 s between seeing lightning and hearing thunder, because light travels much
faster than sound (about 330 m/s in air).