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2 k U =0: (16.77)

2

dz c(z) ;

It is important to note at this point that the frequency ! is a xed constant, and that

according to (16.76) the variable k is an integration variable that can be anything. For

this reason one should at this point not use a relation k = !=c(z) because k can still be

anything.

c0

z=0

c1

z=H

c0

Figure 16.5: Geometry of the model of a single layer sandwiched between two homogeneous

half-spaces.

Now consider the special case of the model shown in gure 16.5. We require that the

waves outside the layer move away from the layer.

Problem b: Show that this implies that the solution for z < 0 is given by A exp (;ik0 z)

and the solution for z > H is given by B exp (+ik0 z) where A and B are unknown

integration constants and where k0 is given by

s

!2 k2 :

k0 = c2 (16.78)

;

0

Problem c: Show that within the layer the wave- eld is given by C cos k1 z + D sin k1 z

with C and D integration constants and k1 is given by

s2

k1 = !2 k2 : (16.79)

c1 ;

The solution in the three regions of space therefore takes the following form:

8

> A exp (;ik0z) for z < 0

<

U(k z) = > C cos k1 z + D sin k1 z for 0 < z < H (16.80)

: B exp (+ik0 z) for z > H

CHAPTER 16. NORMAL MODES

232

We now have the general form of the solution within the layer and the two half-spaces

on both sides of the layer. Boundary conditions are needed to nd the integration constants

A, B, C and D. For this system both U and dU=dz are continuous at z = 0 and z = H.

Problem d: Use the results of problem b and problem c to show that these require-

ment impose the following constraints on the integration constants:

A C=0

;

ik0 A + k1 D = 0 (16.81)

ik0 H + C cos k1 H + D sin k1 H = 0

e

;B

ik0 H + k1 C cos k1 H k1 D sin k1 H = 0 :

ik0 Be ;

This is a linear system of four equations for the four unknowns A, B, C and D. Note that

this is a homogeneous system of equations, because the right hand side vanishes. Such a

homogeneous system of equations only has nonzero solutions when the determinant of the

system of equations vanishes.

Problem e: Show that this requirement leads to the following condition:

k

tan k1 H = k2 +0k21 (16.82)

;2ik

1 o

This equation is implicitly an equation for the wave-number k, because according to

(16.78) and (16.79) both k0 and k1 are a function of the wave-number k. Equation (16.82)

implies that the system can only support waves when the wave-number k is such that

expression (16.82) is satis ed. The system does strictly speaking not have normal modes,

because the waves propagate in the x-direction. However, in the z-direction the waves only

\ t" in the layer for very speci c values of the wave-number k. These waves are called

\guided waves" because they propagate along the layer with a well-de ned phase velocity

that follows from the relation c(!) = !=k. Be careful not to confuse this phase velocity

c(!) with the velocities c1 and c0 in the layer and the half-spaces outside the layer. At

this point we do not know yet what the phase velocities of the guided waves are.

The phase velocity follows from expression (16.82) because this expression is implicitly

an equation for the wave-number k. At this point we consider the case of a low-velocity

layer, i.e. we assume that c1 < c0 . In this case 1=c0 < 1=c1 . We will look for guided waves

with a wave-number in the following interval: !=c0 < k < !=c1 .

Problem f: Show that in that case k1 is real and that k0 is purely imaginary. Write

k0 = i 0 and show that s

k2 !2 :

2

0= (16.83)

c0

;

Problem g: Show that the solution decays exponentially away from the low-velocity

channel both in the half-space z < 0 and the half-space z > H.

The fact that the waves decay exponentially with the distance to the plate means that the

guided waves are trapped near the low-velocity layer. Waves that decay exponentially are

called evanescent waves.

16.8. GUIDED WAVES IN A LOW VELOCITY CHANNEL 233

Problem h: Use (16.82) to show that the wave-number of the guided waves satis es the

following relation

s s

2 2

2;! !

s 2 k c2 c2 k2

!2

;

0 1

k2 H

tan = (16.84)

!2 c12 ; c12

c2 ;

1

1 0

For a xed value of ! this expression constitutes a constraint on the wave-number k of

the guided waves. Unfortunately, it is not possible to solve this equation for k in closed

form. Such an equation is called a transcendental equation.

Problem j: Make a sketch of both the left hand side and the right hand side of expres-

sion (16.84) as a function of k. Show that the two curves have a nite number of

intersection points.

These intersection points correspond to the k-values of the guided waves. The correspond-

ing phase velocity c = !=k in general depends on the frequency !. This means that these

guided waves are dispersive, which means that the di erent frequency components travel

with a di erent phase velocity. It is for this reason that expression (16.82) is called the

dispersion relation.

Dispersive waves occur in many di erent situations. When electromagnetic waves

propagate between plates or in a layered structure, guided waves result 31]. The atmo-

sphere, and most importantly the ionosphere is an excellent waveguide for electromag-

netic waves 27]. This leads to a large variety of electromagnetic guided waves in the

upper atmosphere with exotic names such as \pearls", \whistlers", \tweaks", \hydro-

magnetic howling" and \serpentine emissions" colorful names associated with the sounds

these phenomena would make if they were audible, or with the patterns they generate in

frequency-time diagrams. Guided waves play a crucial role in telecommunication, because

light propagates through optical bers as guided waves 36]. The fact that these waves are

guided prohibits the light to propagate out of the ber, this allows for the transmission

of light signals over extremely large distances. In the Earth the wave velocity increases

rapidly with depth. Elastic waves can be guided near the Earth's surface and the di erent

modes are called \Rayleigh waves" and \Love waves" 2]. These surface waves in the Earth

are a prime tool for mapping the shear-velocity within the Earth 56].

Since the surface waves in the Earth are trapped near the Earth's surface, they propa-

gate e ectively in two dimensions rather than in three dimensions. The surface waves

therefore su er less from geometrical spreading than the body waves that propagate

through the interior of the Earth. For this reason, it is the surface waves that do most

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