GG304 Lecture 18
1
LECTURE 18: SEISMIC WAVES
Seismic waves transmit energy through a medium by means of elastic
displacements without transfer of mass.
A compressional wave
(longitudinal wave) involves
deformations that are oriented in the
direction of propagation of the wave.
If the passage of a wave through position x produces a force Fx and a
displacement u in the x-direction, then we can write:
∂F
∂σ
dFx = x dx = xx Ax dx
if the wave passes through an area Ax.
∂x
∂x
" ∂2u %
Newton’s laws yields: Fx = max = ( ρ Ax dx ) $ 2 '
# ∂t &
The stress-strain relation (in 1-dimension) yields: σ xx = E ε xx = E
Combining these yields:
2
∂2u
2 ∂ u
=V
∂t 2
∂x 2
where
V=
∂u
∂x
E
ρ
This is the equation for a 1-dimensional wave propagating in the x-direction with
speed V. However, in 3-dimensions the material can deform perpendicular to the
propagation direction (if Poisson’s ratio is nonzero). In this case the wave
propagates as a series of dilations and compressions. Consideration of these
volume changes leads to the seismic wave equation:
2
K + 43 µ
∂2θ
λ + 2µ
2 ∂ θ
=
α
where
α=
=
2
2
∂t
∂x
ρ
ρ
is the wave speed.
These are the fastest seismic waves, and are called P-waves (or primary
waves ) because they are the first seismic waves to arrive after an earthquake.
In liquids, μ=0 but K≠0, so p-waves are transmitted with a speed α = K ρ .
Clint Conrad
18-1
University of Hawaii
GG304 Lecture 18
2
A transverse wave (shear
wave) involves deformations
that are oriented
perpendicular to the wave’s
propagation direction.
If the passage of a wave through position x produces a force Fz and a
displacement w in the z-direction, then we can write:
∂F
∂σ
dFz = z dx = xz Ax dx
if the wave passes through an area Ax.
∂x
∂x
" ∂2w %
As before, Newton’s laws yield: Fx = max = ( ρ Ax dx ) $ 2 '
# ∂t &
" ∂w ∂u %
+ '
Here the stress-strain relation (in 1-dimension) yields: σ xz = 2µε xz = µ $
# ∂x ∂z &
∂w
Because the u, displacement in the x-direction, is zero, then σ xz = µ
∂x
2
2
∂w
∂w
µ
= β2 2
Combining these yields:
where
β=
2
∂t
∂x
ρ
This is the equation for a shear wave propagating the x-direction at speed β.
Note that shear waves cannot propagate through liquids or gasses because the
shear modulus μ is zero for these materials.
4
K
Note also that α 2 − β 2 =
and thus α is always greater than β. The transverse
3
ρ
wave is therefore called the secondary wave , or S-wave , because it always
arrives later than the primary wave.
The secondary wave has two perpendicular components: the component with
motion in the vertical (z-direction) is called the SV-wave and the component
with motion in the horizontal (y-direction) is called the SH-wave .
Clint Conrad
18-2
University of Hawaii
GG304 Lecture 18
3
" " x t %%
The solution to the seismic wave equation is: u = A sin $ 2π $ − '' where
# # λ T &&
A is amplitude, λ is wavelength and T=1/f is the period of the seismic wave.
We can write the solution as:
u = A sin (kx − ωt ) = A sin (k ( x − ct )) where:
2π
is the wave number
ω = 2π f is angular frequency (f is frequency)
λ
ω
c = λf = is the phase velocity (α for p-waves or β for s-waves).
k
k=
∂u
= −ω A cos (kx − ωt )
∂t
1
1
The kinetic energy density of the wave is: I = ρv p2 = ρω 2 A 2 cos2 (kx − ωt )
2
2
1
Averaged over a harmonic cycle, the total energy density is I avg = ρω 2 A 2
2
Seismic energy attenuates (decreases in amplitude) as it radiates away from the
E
source. This geometric attenuation scales as I(r ) =
2π r 2
Energy absorption by rock anelasticity causes damping of seismic waves. The
2π
ΔE
=−
fractional loss of energy per cycle is
where Q is the quality factor.
Q
E
" π r%
Amplitude decreases as A = A0 exp $ −
'.
# Q λ&
The vibrational speed of particles in the wave is: v p =
Surface waves travel along a free surface.
Rayleigh waves (LR) combine P- and SVwaves, and travel with speed VLR = 0.9194 β .
Particles shallower than ~0.4λ participate in
the wave, moving in a vertical oval.
Love waves (LQ) involve the reflection of
SH-waves within a layer over a halfspace, and produce horizontal particle
motions as they travel with a speed βlayer <VLQ < βhalfspace . Shorter λ waves travel
more slowly (closer to βlayer ).
Clint Conrad
18-3
University of Hawaii