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displacement at a seismic station in Naroch (Belarus) after an earthquake at Jan-Mayen

island. Around t = 300s and t = 520s impulsive waves arrive, these are the body waves

that travel through the interior of the Earth. The wave with the largest amplitude that

arrives between t = 650s and t = 900s is the surface wave that is guided along the Earth's

surface. Note that the waves that arrive around t = 700s have a lower frequency con-

tent than the waves that arrive later around t = 850s. This is due to the fact that the

CHAPTER 16. NORMAL MODES

234

Figure 16.6: Vertical component of the ground motion at a seismic station in Naroch

(Belarus) after an earthquake at Jan-Mayen island. This station is part of the Network of

Autonomously Recording Seismographs (NARS) which is operated by Utrecht University.

group velocity of the low-frequency components of the surface wave is higher than the

group-velocity of the high-frequency components. Hence it is ultimately the dispersion of

the Rayleigh waves that causes the change in the apparent frequency of the surface wave

arrival.

16.9 Leaky modes

The guided waves in the previous section decay exponentially with the distance to the

low-velocity layer. Intuitively, the fact that the waves are con ned to a region near a

low-velocity layer can be understood as follows. Waves are refracted from regions of a

high velocity to a region of low velocity. This means that the waves that stray out of the

low-velocity channel are refracted back in the channel. E ectively this traps the waves

near the vicinity of the channel. This explanation suggest that for a high-velocity channel

the waves are refracted away from the channel. The resulting wave pattern will then

correspond to waves that preferentially move away from the high velocity layer. For this

reason we consider in this section the waves that propagate through the system shown in

gure 16.5 but we will consider the case of a high-velocity layer where c1 > c0 .

In this case, 1=c1 < 1=c0 , and we will rst consider waves with a wave-number that is

con ned to the following interval: !=c1 < k < !=c0 .

16.9. LEAKY MODES 235

Problem a: Show that in this case the wave-number k1 is imaginary and that it can be

written as k1 = i 1 , with s

k2 !2

2

1= (16.85)

c1

;

and show that the dispersion relation (16.82) is given by:

tan i 1 H = 2 0 k1 : (16.86)

;2k

2

1 o;

Problem b: Use the relation cos x = ;eix + e;ix =2 and the related expression for sin x

to rewrite the dispersion relation (16.86) in the following form:

01

i tanh 1 H = : (16.87)

;2k

2 ; ko

2

1

In this expression all quantities are real when k is real. The factor i in the left hand side

implies that this equation cannot be satis ed for real values of k. The only way in which the

dispersion relation (16.87) can be satis ed is that k is complex. What does it mean that the

wave-number is complex? Suppose that the dispersion relation is satis ed for a complex

wave-number k = kr +iki , with kr and ki the real and imaginary part. In the time domain

a solution behaves for a xed frequency as U(k z) exp i (kx !t). This means that for

complex values of the wave-number the solution behaves as U(k z)e;ki x exp i (kr x !t).

;

;

This is a wave that propagates in the x-direction with phase velocity c = !=kr and that

decays exponentially with the propagation distance x.

The exponential decay of the wave with the propagation distance x is due to the

fact that the wave energy refracts out of the high-velocity layer. A di erent way of

understanding this exponential decay is to consider the character of the wave- eld outside

the layer.

Problem c: Show that in the two half-spaces outside the high-velocity layer the waves

propagate away from the layer. Hint: analyze the wave-number k0 in the half-spaces

and consider the corresponding solution in these regions.

This means that wave energy is continuously radiated away from the high-velocity layer.

The exponential decay of the mode with propagation distance x is thus due to the fact

that wave energy continuously leaks out of the layer. For this reason one speaks of leaky

modes 64]. In the Earth a well-observed leaky mode is the S-PL wave. This is a mode

where a transverse propagating wave in the mantle is coupled to a wave that is trapped

in the Earth's crust.

In general there is no simple way to nd the complex wave-number k for which the

dispersion relation (16.87) is satis ed. However, the presence of leaky modes can be seen

in gure 16.7 where the following function is shown in the complex plane:

F(k) 1= i tanh H + 2k0 1 : (16.88)

1 2 ; k2

1 o

Problem d: Show that this function is in nite for the k-values that correspond to a leaky

mode.

CHAPTER 16. NORMAL MODES

236

Figure 16.7: Contour diagram of the function F(k) for a high-velocity layer with velocity

c1 = 8:4km=s and a thickness H = 15km that is embedded between two halfspaces with

velocity c0 = 8km=s, for waves with a frequency of 5Hz. The horizontal axis is given by

kr =! and the vertical axis by ki.

The function F(k) in gure 16.7 is computed for a high-velocity layer with a thickness of

15km and a velocity of 8:4km=s that is embedded between two half-spaces with a velocity

of 8km=s. The frequency of the wave is 5Hz. In this gure, the horizontal axis is given by

(p) = kr =! while the vertical axis is given by (!p) = ki . The quantity p is called

<e =m

the slowness because (p) = kr =! = 1=c(!). The leaky modes show up in gure 16.7 as

<e

a number of localized singularities of the function F(k).

Problem e: What is the propagation distance over which the amplitude of the mode

with the lowest phase velocity decays with a factor 1=e?

Leaky modes have been used by Gubbins and Snieder 26] to analyze waves that have

propagated along a subduction zone. (A subduction zone is a plate in the Earth that

slides downward in the mantle.) By a fortuitous geometry, compressive wave that are

excited by earthquakes in the Tonga-Kermadec region that travel to a seismic station in

Wellington (New Zealand) propagate for a large distance through the Tonga-Kermadec

subduction zone. At the station in Wellington, a high-frequency wave arrives before the

main compressive wave. This can be seen in gure 16.8 where such a seismogram is shown

band-pass ltered at di erent frequencies. It can clearly be seen that the waves with

a frequency around 6Hz arrive before the waves with a frequency around 1Hz. This

observation can be explained by the propagation of a leaky mode through the subduction

zone. The physical reason that the high-frequency components arrive before the lower

frequency components is that the high-frequency waves \ t" in the high-velocity layer

in the subducting plate, whereas the lower frequency components do not t in the high-

velocity layer and are more in uenced by the slower material outside the high-velocity

layer. Of course, the energy leaks out of the high-velocity layer so that this arrival is very

16.10. RADIATION DAMPING 237

Figure 16.8: Seismic waves recorded in Wellington after an earthquake in the Tonga-

Kermadec subduction zone, from Ref. 26]. The di erent traces correspond to the waves

band-pass ltered with a center frequency indicated at each trace. The horizontal axis

gives the time since the earthquake in seconds.

weak. From the data it could be inferred that in the subduction zone a high-velocity layer

is present with a thickness between 6 and 10 km 26].

16.10 Radiation damping

Up to this point we have considered systems that are freely oscillating. When such systems

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