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Problem f: Find the eigen-frequencies of the four modes of the drum with the lowest

frequencies and make a sketch of the associated wave standing wave of the drum.

Problem g: Compute the separation between the di erent zero crossing for a xed value

of m. To which number does this separation converge for the zero crossings at large

values of x?

Using the results of this section it follows that the modes of the drum are given by

(m)

unm(r ') = Jm (kn r)eim' : (16.19)

CHAPTER 16. NORMAL MODES

214

Problem h: Let us rst consider the '-dependence of these modes. Show that when

one follows the mode unm (r ') along a complete circle around the origin that one

encounters exactly m oscillations of that mode.

The shape of the Bessel function is more di cult to see than the properties of the functions

eim' . As shown in section 9.7 of Butkov 14] these functions satisfy a large number of

properties that include recursion relations and series expansions. However, at this point

the following facts are most important:

The Bessel functions Jm (x) are oscillatory functions that decay with distance, in a

sense they behave as decaying standing waves We will return to this issue in section

16.5.

The Bessel functions satisfy an orthogonality relation similar to the orthogonality

relation (16.6) for the modes of the string. This orthogonality relation is treated in

more detail in section 16.4.

16.3 The normal modes of a sphere

In this section we consider the normal modes of a spherical surface with radius R. We only

consider the modes that are associated with the waves that propagate along the surface,

hence we do not consider wave motion in the interior of the sphere. The modes satisfy the

wave equation (16.1). Since the waves propagate on the spherical surface, they are only a

function of the angles and ' that are used in spherical coordinates: u = u( '). Using

the expression of the Laplacian in spherical coordinates the wave equation (16.1) is then

given by ( 2u )

1 @ sin @u + 1 @

1 2

2 @'2 + k u = 0 : (16.20)

2 sin @

R @ sin

Again, we will seek a solution by applying separation of variables by writing the solution

in a form similar to (16.9):

u( ') = F( )G(') : (16.21)

Problem a: Insert this in (16.20) and apply separation of variables to show that F( )

satis es the following di erential equation:

sin dd sin dF + k2 R2 sin2 F =0 (16.22)

d ;

and that G(') satis es (16.12), where the unknown constant does not depend on

or '.

To make further progress we have to apply boundary conditions. Just as with the

drum of section 16.2 the system is invariant when a rotation over 2 is applied: u( ') =

u( ' + 2 ). This means that G(') satis es the same di erential equation (16.12) as for

the case of the drum and satis es the same periodic boundary condition (16.14). The

16.3. THE NORMAL MODES OF A SPHERE 215

2,

solution is therefore given by G(') = eim' and the separation constant satis es = ;m

with m an integer. Using this, the di erential equation for F( ) can be written as:

!

1 d sin dF + k2 R2 m2 F =0: (16.23)

sin2

sin d d ;

Before we continue let us compare this equation with expression (16.11) for the modes of

the drum that we can rewrite as

!

1 d r dF + k2 m2 F = 0 (16.11) again

r dr dr r2

;

Note that these equations are identical when we compare r in (16.11) with sin in (16.23).

There is a good reason for this. Suppose the we have a source in the middle of the drum.

In that case the variable r measures the distance of a point on the drum to the source.

This can be compared with the case of waves on a spherical surface that are excited by

a source at the north-pole. In that case, sin is a measure of the distance of a point to

the source point. The only di erence is that sin enters the equation rather than the

true angular distance . This is a consequence of the fact that the surface is curved, this

curvature leaves an imprint on the di erential equation that the modes satisfy.

Problem b: The di erential equation (16.11) was reduced in section 16.2 to the Bessel

equation by changing to a new variable x = kr. De ne a new variable

x cos (16.24)

and show that the di erential equation for F is given by

!

d 1 x2 dF + k2 R2 m2 F = 0 : (16.25)

dx dx 1 x2

; ;

;

The solution of this di erential equation is given by the associated Legendre functions

Plm (x). These functions are described in great detail in section 9.8 of Butkov 14]. In fact,

just as the Bessel equation, the di erential equation (16.25) has a solution that is regular

as well a solution Qm (x) that is singular at the point x = 1 where = 0. However, since

l

the modes are nite everywhere, they are given by the regular solution Plm (x) only.

The wave-number k is related to frequency by the relation k = !=c. At this point it is

not clear what k is, hence the eigen-frequencies of the spherical surface are not yet known.

It is shown in section 9.8 of Butkov 14] that:

The associated Legendre functions are only nite when the wave-number satis es

k2 R2 = l(l + 1) (16.26)

where l is a positive integer. Using this in (16.25) implies that the associated Leg-

endre functions satisfy the following di erential equation:

!

1 d sin dPlm (cos ) + l (l + 1) m2 Plm (cos ) = 0 : (16.27)

sin2

sin d d ;

CHAPTER 16. NORMAL MODES

216

Seen as a function of x (= cos ) this is equivalent to the following di erential

equation

!

d 1 x2 dPlm (x) + l (l + 1) m2 P m (x) = 0 : (16.28)

1 x2 l

dx dx

; ;

;

The integer l must be larger or equal than the absolute value of the angular order

m.

Problem c: Show that the last condition can also be written as:

m l: (16.29)

;l

Problem d: Derive that the eigen-frequencies of the modes are given by

p

l(l + 1)R

!l = : (16.30)

c

It is interesting to compare this result with the eigen-frequencies (16.7) of the string.

The eigen-frequencies of the string all have the same spacing in frequency, but the eigen-

frequencies of the spherical surface are not spaced at the same interval. In musical jargon

one would say that the overtones of a string are harmonious, this means that the eigen-

frequencies of the overtones are multiples of the eigen-frequency of the ground tone. In

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