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section leaky modes were introduced. In such a system, energy is radiated away, which

leads to an exponentially of waves that propagate through the system. In a similar way,

a system that has normal modes when it is isolated from its surroundings, can display

damped oscillations when it is coupled to the external world.

As a simple prototype of such a system, consider a mass m that can move in the

z-direction which is coupled to a spring with spring constant . The mass is attached to

a string that is under a tension T and which has a mass per unit length. The system

is shown in gure 16.9. The total force acting on the mass is the sum of the force z ;

exerted by the spring and the force Fs that is generated by the string:

mz + z = Fs (16.89)

where z denotes the vertical displacement of the mass. The motion of the waves that

CHAPTER 16. NORMAL MODES

238

z

x

Figure 16.9: Geometry of an oscillating mass that is coupled to a spring.

propagate in the string is given by the wave equation:

1u =0

u (16.90)

xx ; c2 tt

where u is the displacement of the string in the vertical direction and c is given by

s

c= T : (16.91)

Let us rst consider the case that no external force is present and that the mass is not

coupled to the spring.

Problem a: Show that in that case the equation of motion is given by

2

z + !0 z = 0 (16.92)

with !0 given by r

!0 = m : (16.93)

One can say that the mass that is not coupled to the string has one free oscillation with

angular frequency !0 . The fact that the system has one free oscillation is a consequence

from the fact that this mass can move only in the vertical direction, hence it has only one

degree of freedom.

Before we couple the mass to the string let us rst analyze the wave-motion in the

string in the absence of the mass.

Problem b: Show that for any function f(t x ) satis es the wave equation (16.90).

c ;

Show that this function describes a wave that move in the positive x-direction with

velocity c.

Problem c: Show that any function g(t + x ) satis es the wave equation (16.90). Show

c

that this function describes a wave that move in the negative x-direction with velocity

c.

The general solution is a superposition of the rightward and leftward moving waves:

u(x t) = f(t x ) + g(t + x ) : (16.94)

c c

;

16.10. RADIATION DAMPING 239

This general solution is called the d'Alembert solution.

Now we want to describe the motion of the coupled system. Let us rst assume

that the mass oscillates with a prescribed displacement z(t) and nd the waves that this

displacement generates in the string. We will consider here the situation that there are no

waves moving towards the mass. This means that to the right of the mass the waves can

only move rightward and to the left of the mass there are only leftward moving waves.

Problem d: Show that this radiation condition implies that the waves in the string are

given by: (

f(t x ) for x > 0

u(x t) = g(t + x )

c (16.95)

;

for x < 0

c

Problem e: At x = 0 the displacement of the mass is the same as the displacement of

the string. Show that this implies that f(t) = g(t) = z(t), so that

( x

z(t c for x > 0

u(x t) = z(t + x ) (16.96)

;

for x < 0

c)

T

F = T sin Ï†

Ï†

+

Figure 16.10: Sketch of the force exerted by the spring on the mass.

Now we have solved the problem of nding the wave motion in the string given the

motion of the mass. To complete our description of the system we also need to specify

how the motion of the string a ects the mass. In other words, we need to nd the force Fs

in (16.89) given the motion of the string. This force can be derived from gure 16.10. The

vertical component F+ of the force acting on the mass from the right side of the string is

given by F+ = T sin ', where T is the tension in the string. When the motion in the spring

is su ciently weak we can approximate: F+ = T sin ' T' T tan ' Tux (x = 0+ t).

In the last identity we used that the derivative ux (x = 0+ t) gives the slope of the string

on the right of the point x = 0.

Problem f: Use a similar reasoning to determine the force acting in the mass from the

left part of the spring and show that the net force acting on the spring is given by

;

Fs(t) = T ux(x = 0+ t) ux(x = 0; t) (16.97)

;

where ux (x = 0; t) is the x-derivative of the displacement in the string just to the

left of the mass.

Problem g: Show that this expression implies that the net force that acts on the mass

is equal to the kink in the spring at the location of the mass.

You may not feel comfortable with the fact the we used the approximation of a small angle

' in the derivation of (16.97). However, keep in mind that the wave equation (16.90) is

derived using the same approximation and that this wave equation therefore is only valid

for small displacements of the string.

At this point we have assembled all the ingredients for solving the coupled problem.

CHAPTER 16. NORMAL MODES

240

Problem h: Use (16.96) and (16.97) to derive that the force exerted by the spring on the

mass is given by

Fs(t) = 2T z_ (16.98)

c

;

and that the motion of the mass is therefore given by:

z + 2T z_ + !2 z = 0 : (16.99)

0

mc

It is interesting to compare this expression for the motion of the mass that is coupled

to the string with the equation of motion (16.92) for the mass that is not coupled to the

2T _

string. The string leads to a term mc z in the equation of motion that damps the motion

of the mass. How can we explain this damping physically? When the mass moves, the

string moves with it at location x = 0. Any motion in the string at that point will start

propagating in the string. This means that the string radiates wave energy away from the

mass whenever the mass moves. Since energy is conserved, the energy that is radiated

in the string must be withdrawn from the energy of the moving mass. This means that

the mass looses energy whenever it moves this e ect is described by the damping term

in equation (16.99). This damping process is called radiation damping, because it is the

radiation of waves that damps the motion of the mass.

The system described in this section is extremely simple. However, is does contain

the essential physics of radiation damping. Many systems in physics that display normal

modes are not quite isolated from their surroundings. Interactions of the system with

their surrounding often lead to the radiation of energy, and hence to a damping of the

oscillation of the system.

One example of such a system is an atom in an excited state. In the absence of external

in uences such an atom will not emit any light and decay. However, when such an atom

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