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On the sphere the situation is completely analogous. The modes can be written according

1

to (16.58) as standing waves cos l + 2 =

p

;eix + e;ix =2 the (2m + 1) 4 also sin seen theasphere. However,

on

;

using the relation cos x = modes can be as superposition of

travelling waves ei(l+ 2 ) = sin and e;i(l+ 2 ) = sin on the sphere.

p p

1 1

Problem h: Explain why the rst wave travels away from the north pole while the second

wave travels towards the north pole.

Problem i: Suppose the waves are excited by a source at the north pole. According to

the last problem the motion of the sphere can alternatively be seen as a superpo-

sition of standing waves or of travelling waves. The travelling wave ei(l+ 2 ) = sin

p

1

moves 1away from the source. Explain physically why there is also a travelling wave

e;i(l+ 2 ) = sin moving towards the source.

p

These results imply that the motion of the Earth can either be seen as a superposition

of normal modes, or of a superposition of waves that travel along the Earth's surface in

opposite directions. The waves that travel along the Earth's surface are called surface

waves. The relation between normal modes and surface waves is treated in more detail by

Dahlen 17] and by Snieder and Nolet 57].

16.7 Normal modes and the Green's function

In section (10.7) we analyzed the normal modes of a system of three coupled masses. This

system had three normal modes, and each mode could be characterized with a vector x

with the displacement of the three masses. The response of the system to a force F acting

on the three masses with time dependence exp (;i!t) was derived to be:

1 X v(n) (^(n) F)

3^ v

x =m (10.83) again :

n=1 (!n ! )

2 2

;

This means that the Green's function of this system is given by the following dyad:

1 X v(n) v(n)T :

3^^

G = 2 m (!2 !2) (16.60)

n=1 n ;

The factor 1=2 is due to the fact that a delta function f(t) = (t) force in the time-domain

corresponds with the Fourier transform (11.43) to F(!) = 1=2 in the frequency-domain.

16.7. NORMAL MODES AND THE GREEN'S FUNCTION 227

In this section we derive the Green's function for a very general oscillating system that

can be continuous. An important example is the Earth, which a body that has well-

de ned normal modes and where the displacement is a continuous functions of the space

coordinates.

We consider a system that satis es the following equation of motion:

u + Hu = F : (16.61)

The eld u can either be a scalar eld or a vector eld. The operator H is at this point

very general, the only requirement that we impose is that this operator is Hermitian, this

means that we require that

(f Hg) = (Hf g) (16.62)

R f g dV . In the frequency domain, the

where the inner product of is de ned as (f h)

equation of motion is given by

!2 u + Hu = F(!) : (16.63)

;

Let the normal modes of the system be denoted by u(n) , the normal modes describe

the oscillations of the system in the absence of any external force. The normal modes

therefore satisfy the following expression

Hu(n) = !n u(n)

2 (16.64)

where !n is the eigen-frequency of this mode.

Problem a: Take the inner product of this expression with a mode u(m) , use that H is

Hermitian to derive that

!n !m2 u(m) u(n) = 0 :

2 (16.65)

;

Note the resemblance of this expression with (16.41) for the modes of a system that obeys

the Helmholtz equation.

Problem b: Just as in section 16.4 one can show that the eigen-frequencies are real by

setting m = n, and one can derive that di erent modes are orthogonal with respect

to the following inner product:

u(m) u(n) = for !m = !n : (16.66)

nm 6

Give a proof of this orthogonality relation.

Note the presence of the density term in this inner product.

Let us now return to the inhomogeneous problem (16.63) where an external force F(!)

is present. Assuming that the normal modes form a complete set, the response to this

force can be written as a sum of normal modes:

X

cn u(n)

u= (16.67)

n

where the cn are unknown coe cients.

CHAPTER 16. NORMAL MODES

228

Problem c: Find these coe cients by inserting (16.67) in the equation of motion (16.63)

and by taking the inner product of the result with a mode u(m) to derive that

u(m) F

cm = !2 !2 : (16.68)

m ;

This means that the response of the system can be written as:

X u(n) u(n) F

u= : (16.69)

!n !2

2

n ;

Note the resemblance of this expression with equation (10.83) for a system of three masses.

The main di erence is that the derivation of this section is valid as well for continuous

vibrating systems such as the Earth.

It is instructive to re-write this expression taking the dependence of the space coordi-

nates explicitly into account:

X u(n) (r) R u (n) (r0 )F (r0 )dV 0

u(r) = : (16.70)

!n !2

2

n ;

It follows from this expression that the Green's function is given by

0 !) = 1 X u(n)(r)u (n) (r0 ) :

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