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;

t = r=c, but it is nonzero everywhere within this wave front.. This means that in two

dimensions an impulsive input leads to a sound response that is of in nite duration. One

can therefore say that:

Any word spoken in two dimensions will reverberate forever (albeit weakly).

The approach we have taken is to compute the Green's function in two dimension

is interesting in that we solved the problem rst in a higher dimension and retrieved

CHAPTER 15. GREEN'S FUNCTIONS, EXAMPLES

206

the solution by integrating over one space dimension. Note that for this trick it is not

necessary that this higher dimensional space indeed exists! (Although in this case it does.)

Remember that we took this approach because we did not want to evaluate the Fourier

transform of a Hankel function. We can also turn this around the Green's function (15.49)

can be used to determine the Fourier transform of the Hankel function.

Problem h: Show that the Fourier transform of the Hankel function is given by:

(0

Z 1 (1) for q < 1

;iqxdx = 2 1

H (x)e (15.50)

for q > 1

;1 0 i q2 ;1 p

Let us continue with the Green's function of the wave equation in one dimension in

the time domain.

Problem i: Use the Green's function for one dimension of the last section to show that

in the time domain

Z

1D (x t) = ic 1 1 e;i!(t;jxj=c) d! :

G (15.51)

4 ;1 !

;

This integral resembles the integral used for the calculation of the Green's function in

three dimensions. The only di erence is the term 1=! in the integrand, because of this

term we cannot immediately evaluate the integral. However, the 1=! term can be removed

by di erentiating expression (15.51) with respect to time, and the remaining integral can

be evaluated analytically.

Problem j: Show that

@G1D (x t) = c t : (15.52)

jxj

@t c

2 ;

Problem k: This expression can be integrated but one condition is needed to specify the

integration constant that appears. We will use here that at t = the Green's

;1

function vanishes. Show that with this condition the Green's function is given by:

(

for t < =c

0

G1D (x t) = (15.53)

jxj

c=2 for t > =c

jxj

Just as in two dimensions the solution is nonzero everywhere within the expanding wave

front and not only on the wave front = ct such as in three dimensions. However, there

jxj

is an important di erence in two dimensions the solution changes for all times with time

whereas in one dimension the solution is constant except for t = =c. Humans cannot

jxj

detect a static change in pressure (did you ever hear something when you drove in the

mountains?), therefore a one-dimensional human will only hear a sound at t = =c but jxj

not at later times.

In order to appreciate the di erence in the sound propagation in 1, 2 and 3 space

dimensions the Green's functions for the di erent dimensions is shown in gure (15.3).

Note the dramatic change in the response for di erent numbers of dimensions. This change

in the properties of the Green's function with change in dimension has been used somewhat

15.4. THE WAVE EQUATION IN 1,2,3 DIMENSIONS 207

jokingly by Morley 41] to give \a simple proof that the world is three dimensional." When

you have worked through the sections (15.1) and (15.2) you have learned that both for the

heat equation and the Schrodinger equation the solution does not depend fundamentally

on the number of dimensions. This is in stark contrast with the solutions of the wave

equation that depend critically on the number of dimensions.

CHAPTER 15. GREEN'S FUNCTIONS, EXAMPLES

208

Chapter 16

Normal modes

Many physical systems have the property that they can carry out oscillations only at

certain speci c frequencies. As a child (and hopefully also as an adult) you will have

discovered that a swing on a playground will move only with a very speci c natural

period, and that the force that pushes the swing is only e ective when the period of the

force matches the period of the swing. The patterns of motion at which a system oscillates

are called the normal modes of the system. A swing may has one normal mode, but you

have seen in section 10.7 that a simple model of a tri-atomic molecule has three normal

modes. An example of a normal mode of a system is shown in gure 16.1. Shown is the

Figure 16.1: Sand on a metal plate that is driven by an oscillator at a frequency that

corresponds to one of the eigen-frequencies of the plate. This gure was prepared by John

Scales at the Colorado School of Mines.

pattern of oscillation of a metal plate that is driven by an oscillator at a xed frequency.

The screw in the middle of the plate shows the point at which the force on the plate is

applied. Sand is sprinkled on the plate. When the frequency of the external force is equal

to the frequency of a normal mode of the plate, the motion of the plate is given by the

motion that corresponds to that speci c normal mode. Such a pattern of oscillation has

nodal lines where the motion vanishes. These nodal lines are visible because the sand on

209

CHAPTER 16. NORMAL MODES

210

the plate collects at the these lines.

In this chapter, the normal modes of a variety of systems are analyzed. Normal modes

play an important role in a variety of applications because the eigen-frequencies of normal

modes provide important information of physical systems. Examples are the normal modes

the Earth that provide information about the internal structure of our planet, or the

spectral lines of light emitted by atoms that have led to the advent of quantum mechanics

and its description of the internal structure of atoms. In addition, normal modes are used

in this chapter to introduce some properties of special functions such as Bessel functions

and Legendre functions. This is achieved by analyzing the normal modes of a system in

1, 2 and 3 dimensions in the sections 16.1 through 16.3.

16.1 The normal modes of a string

In this section and the following two sections we assume that the motion of the system is

governed by the Helmholtz equation

2 u + k2 u = 0 : (16.1)

r

In this expression the wave-number k is related to the angular frequency ! by the relation

k=! : (16.2)

c

For simplicity we assume the system to be homogeneous, this means that the velocity c is

constant. This in turn implies that the wave-number k is constant. In the sections 16.1

through 16.3 we consider a body with size R. Since a circle or sphere with radius R has

a diameter 2R we will consider here a string with length 2R in order to be able to make

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