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Problem e: Show that for large values of l the eigen-frequencies of the spherical surface

have an almost equal spacing.

Problem f: The eigen-frequency !l only depends on the order l but not on the degree m.

For each value of l, the angular degree m can according to (16.29) take the values

l 1 l. Show that this implies that for every value of l, there are

+1

;l ;l ;

(2l + 1) modes with the same eigen-frequency.

When di erent modes have the same eigen-frequency one speaks of degenerate modes.

The results we obtained imply that the modes on a spherical surface are given by

Plm (cos )eim'. We used here that the variable x is related to the angle through the

relation (16.24). The modes of the spherical surface are called spherical harmonics. These

eigen-functions are for m 0 given by:

s

(l m)!

Ylm( ') = (;1)m 2l4+ 1 (l + m)! Plm (cos )eim' m 0 : (16.31)

;

For m < 0 the spherical harmonics are de ned by the relation

Ylm ( ') = (;1)m Yl ;m( ') (16.32)

You may wonder where the square-root in front of the associated Legendre function comes

from. One can show that with this numerical factor the spherical harmonics are normalized

when integrated over the sphere:

ZZ 2d = 1 (16.33)

lm j

jY

16.3. THE NORMAL MODES OF A SPHERE 217

RR d denotes an integration over the unit sphere. You have to be aware of the

where

fact that di erent authors use di erent de nitions of the spherical harmonics. For example,

˜

one could de ne the spherical harmonics also as Ylm ( ') = Plm (cos )eim' because the

functions also account for the normal modes of a spherical surface.

Problem g: Show that the modes de ned in this way satisfy RR Ylm 2 d = 4 = (2l + 1)

˜

(l + m)!=(l m)!.

;

This means that the modes de ned in this way are not normalized when integrated over

the sphere. There is no reason why one cannot work with this convention, as long as one

accounts for the fact that in this de nition the modes are not normalized. Throughout

this book we will use the de nition (16.31) for the spherical harmonics. In doing so we

follow the normalization that is used by Edmonds 20].

Just as with the Bessel functions, the associated Legendre functions satisfy recursion

relations and a large number of other properties that are described in detail in section 9.8

of Butkov 14]. The most important properties of the spherical harmonics Ylm ( ') are:

These functions display m oscillations when the angle ' increases with 2 . In other

words, there are m oscillations along one circle of constant latitude.

The associated Legendre functions Plm (cos ) behave like Bessel functions that they

behave like standing waves with an amplitude that decays from the pole. We return

to this issue in section 16.6.

There are l m oscillations between the north pole of the sphere and the south pole

;

of the sphere.

The spherical harmonics are orthogonal for a suitably chosen inner product, this

orthogonality relation is derived in section 16.4.

A last and very important property is that the spherical harmonics are the eigen-functions

of the Laplacian in the sphere.

Problem h: Give a proof of this last property by showing that

2 Ylm ( ') = (l + 1) Ylm ( ') (16.34)

1

r ;l

where the Laplacian on the unit sphere is given by

@ sin @ + 1 @ 2 :

1

2

r1 = (16.35)

sin2 @'2

@ @

sin

This property is in many applications extremely useful, because the action of the Laplacian

on a sphere can be replaced by the much simpler multiplication with the constant (l + 1)

;l

when spherical harmonics are concerned.

CHAPTER 16. NORMAL MODES

218

16.4 Normal modes and orthogonality relations

The normal modes of a physical system often satisfy orthogonality relations when a suit-

ably chosen inner product for the eigen-functions is used. In this section this is illustrated

by studying once again the normal modes of the Helmholtz equation (16.1) for di erent

geometries. In this section we derive rst the general orthogonality relation for these nor-

mal modes. This is then applied to the normal modes of the previous sections to derive

the orthogonality relations for Bessel functions and associated Legendre functions.

Let us consider two normal modes of the Helmholtz equation (16.1), and let these

modes be called up and uq . At this point we leave it open whether the modes are de ned

on a line, on a surface of arbitrary shape or a volume. The integration over the region of

R dN x, where N is the dimension of

space in which the modes are de ned is denoted as

this space. The wave-number of these modes that acts as an eigenvalue in the Helmholtz

equation is de ned by kp and kq respectively. In other words, the modes satisfy the

equations:

2 up + k2 u = 0 (16.36)

p

r

2 uq + k2 uq = 0 : (16.37)

q

r

The subscript p may stand for a single mode index such as in the index n for the wave-

number kn for the modes of a string, or it may stand for a number of indices such as the

indices nm that label the eigen-functions (16.19) of a circular drum.

Problem a: Multiply (16.36) with uq , take the complex conjugate of (16.37) and multiply

the result with up . Subtract the resulting equations and integrate this over the region

of space for which the modes are de ned to show that

Z Z

2 2 kp ; kq 2

2

dN x + uq updN x = 0 :

uq r up ; up r u (16.38)

q

Problem b: Use the theorem of Gauss to derive that

Z I Z

uq 2up dN x = uq dN x

dS; (16.39)

p p

q

r ru ru ru

H dS is over the surface that bounds the body. If you have

where the integral

trouble deriving this, you can consult expression (6.9) of section 6.3 where a similar

result was used for the derivation of the representation theorem for acoustic waves.

Problem c: Use the last result to show that

I Z

dS+ kp kq 2

2 uq updN x = 0 :

uq p ; upruq (16.40)

ru ;

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