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6

2

Problem k: Find an alternative derivation of this orthogonality relation in the literature.

The result you obtained in problem h implies that the spherical harmonics are or-

thogonal when m1 = m2 because of the '-integration, whereas problem i implies that

6

the spherical harmonics are orthogonal when l1 = l2 because of the -integration. This

6

means that the spherical harmonics satisfy the following orthogonality relation:

ZZ

Yl1 m1 ( ')Yl2 m2 ( ')d = : (16.51)

l1 l2 m1 m2

The numerical constant multiplying the delta functions is equal to 1, this is a consequence

of the square-root term in (16.31) that pre-multiplies the associated Legendre functions.

One should be aware of the fact that when a di erent convention is used for the normal-

ization of the spherical harmonics a normalization factor appears in the right hand side of

the orthogonality relation (16.51) of the spherical harmonics.

16.5 Bessel functions are decaying cosines

As we have seen in section 16.2 the modes of the circular drum are given by Jm (kr)eim'

where the Bessel function satis es the di erential equation (16.16) and where k is a wave-

number chosen in such a way that the displacement at the edge of the drum vanishes. We

will show in this section that the waves that propagate through the drum have approxi-

mately a constant wavelength, but that their amplitude decays with the distance to the

center of the drum. The starting point of the analysis is the Bessel equation

!

d2 Jm + 1 dJm + 1 m2 J = 0 (16.16) again:

x2 m

dx2 x dx ;

1

If the terms x dJm and m22 would be absent in (16.16) the Bessel equation would reduce

dx x

to the di erential equation d2 F=dx2 + F = 0 whose solutions are given by a superposition

of cos x and sin x. We therefore can expect the Bessel functions to display an oscillatory

behavior when x is large.

It follows directly from (16.16) that the term m2 =x2 is relatively small for large values

of x, speci cally when x m. However, it is not obvious under which conditions the term

1 dJm

x dx is relatively small. Fortunately this term can be transformed away.

Problem a: Write Jm (x) = x gm (x), insert this in the Bessel equation (16.16), show that

the term with the rst derivative vanishes when = and that the resulting

;1=2

di erential equation for gm (x) is given by

!

d2 gm + 1 m2 1=4 g = 0 : (16.52)

;

m

dx2 x2

;

CHAPTER 16. NORMAL MODES

222

Up to this point we have made no approximation. Although we have transformed the rst

derivative term out of the Bessel equation, we still cannot solve (16.52). However, when

x m the term proportional to 1=x2 in this expression is relatively small. This means

that for large values of x the function gm (x) satis es the following approximate di erential

equation d2 gm =dx2 + gm 0.

Problem b: Show that the solution of this equation is given by gm (x) A cos (x + '),

where A and ' are constants. Also show that this implies that the Bessel function

is approximately given by:

J (x) A cos (x + ') : (16.53)

m x

p

This approximation is obtained from a local analysis of the Bessel equation. Since all values

of the constants A and ' lead to a solution that approximately satis es the di erential

equation (16.52), it is not possible to retrieve the precise values of these constant from the

analysis of this section. An analysis based on the asymptotic evaluation of the integral

representation of the Bessel function 7] shows that:

r2

Jm (x) = x cos x (2m + 1) 4 + O(x;3=2 ) : (16.54)

;

Problem c: As a check on the accuracy of this asymptotic expression let us compare the

zeroes of this approximation with the zeroes of the Bessel functions as given in table

16.1 of section 16.2. In problem g of section 16.2 you found that the separation of

the zero crossings tends to for large values of x. Explain this using the approximate

expression (16.54) How large must x be for the di erent values of the order m so

that the error in the spacing of the zero crossing is less than 0:01?

Physically, expression (16.54) states that Bessel functions behave like standing waves

with a constant wavelength and which decay with distance as 1= kr. (Here it is used

p

that the modes are given by the Bessel functions with argument x = kr.) How can we

explain this decay of the amplitude with distance? First let us note that (16.54) expresses

the Bessel function in a cosine, hence this is a representation of the Bessel function as

;

a standing wave. However, using the relation cos x = eix + e;ix =2 the Bessel function

can be written as two travelling waves that depend on the distance as (exp ikr) = kr

p

and that interfere to give the standing wave pattern of the Bessel function. Now let us

consider a propagating wave A(r) exp (ikr) in two dimensions, in this expression A(r) is an

amplitude that is at this point unknown. The energy varies with the square of the wave-

2 . The energy current therefore also varies as 2.

eld, and thus depends on jA(r)j jA(r)j

Consider an outgoing wave as shown in gure 16.2. The total energy ux through a ring

of radius r is given by the energy current times the circumference of the ring, this means

2 . Since energy is conserved, this total energy ux is

that the ux is equal to 2 r jA(r)j

2 = constant.

the same for all values of r, which means that 2 r jA(r)j

Problem d: Show that this implies that A(r) 1= r . p

This is the same dependence on distance as the 1= x decay of the approximation (16.54)

p

of the Bessel function. This means that the decay of the Bessel function with distance is

dictated by the requirement of energy conservation.

16.6. LEGENDRE FUNCTIONS ARE DECAYING COSINES 223

ikr

A(r)e

r

Figure 16.2: An expanding wavefront with radius r.

16.6 Legendre functions are decaying cosines

The technique used in the previous section for the approximation of the Bessel function

can also be applied to spherical harmonics. We will show in this section that the spherical

harmonics behave asymptotically as standing waves on a sphere with an amplitude decay

that is determined by the condition that energy is conserved. The spherical harmonics

are proportional to the associated Legendre functions with argument cos , the starting

point of our analysis therefore is the di erential equation for Plm (cos ) that was derived

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