Chapter 16

Waves IIn this chapter we will start the discussion on wave phenomena. We will study the following topics:

Types of waves Amplitude, phase, frequency, period, propagation speed of a wave Mechanical waves propagating along a stretched string Wave equation Principle of superposition of waves Wave interference Standing waves, resonance

(16–1)

Waves can be divided into the following two categoriesdepending on the orientation of the disturbance with respect to the wave propagation velocity .

If the disturba

v

Transverse and Longitudinal Waves

nce associated with a particular wave is perpendicular to the wave propagation velocity, this wave is called " " An example is given in the upper figure, which depicts a mechanical wave tha

transverse.

t propagates along a string. The movement of each particle on the string is along the -axis; the wave itselfpropagates along the -axis.

yx

A wave in which the associated disturbance is parallel to the wave propagationvelocity is known as a " " An example of such a wave is given in the lower figure. It is produced by a

longitudinal wave.piston oscillating in a tube filled with

air. The resulting wave involves movement of the air molecules along the axis of the tube, which is also the direction of the wave propagation velocity .v

(16–3)

Consider the transverse wave propagating along the string as shown in the figure. The position ofany point on the string can be described by a function ( , ). Further along in the chapterwe shal

y h x t

l see that function has to have a specificform to describe a wave. One such suitable function is ( , ) sin - .

Sum

h

y x t y kx t

ch a wave, which is described by a sine (or a cosine) function, is known as a" "The various terms that appear in the expression for a harmonic wave are identified in the lower figure.Func

harmonic wave.

0

0 0

tion ( , ) depends on and . There are two ways to visualize it. The first is to "freeze" time (i.e., set ). This is like taking a snapshot of the

wave at : , . The second is to set

y x t x t

t t

t t y y x t

0 0

. In this case , .x x y y x t (16–4)

The is the absolute value of themaximum displacement from the equilibriumposition. The is defined as the argument

of the sine function.The

( , ) sin

is the

m

my

kx

y x t y kx t

t

amplitude

phase

wavelength shortest distance between two repetitions of the wave at a fixed time.

1 1

1 1 1

We fix at 0. We have the condition: ( ,0) ( ,0)

sin sin sin .

Since the sine function is periodic with period 2 2

A is the time it takes (with fixe

2 .

m m m

t t y x y x

y

k

kx y k x y kx k

k

T

period

d ) for the sine function to complete one oscillation.

2We take 0 (0, ) (0, )

sin sin sin 2 .m m m

xx y t y t T

y t y t T y t T TT

(16–5)

In the figure we show two snapshots of a harmonic wavetaken at times and . During the time interval the wave has traveled a distance . The wave speed

. On

t t t tx

xvt

The Speed of a Traveling Wave

e method of finding is to imagine that

we move with the same speed along the -axis. In thiscase it will seem to us that the wave does not change.

v

x

Since ( , ) sin this means that the argument of the sine function

is constant: constant. We take the derivative with respect to :

0 . The speed .

A harmoni

my x t y kx t

kx t tdx dx dxk vdt dt k dt k

c wave that propagates along the -axis is described by the equation

( , ) sin . The function ( , ) describes a general wave that

propagates along the positive -axis. A genem

xy x t y kx t y x t h kx t

x

negative

ral wave that propagates along the

-axis is described by the equati (on , ) .y x t h kx tx negative

vk

(16–6)

Example 2:

Example 3:

Example 4:

Consider a transverse wave propagating along a string, which is described by the equation ( , ) sin - . The transverse velocity

- cos - . At point bot

m

m

y x t y kx tyu y kx t at

Rate of Energy Transmission

h

and are equal to zero. At point both and have maxima. y u b y u

2

2

2

1In general, the kinetic energy of an element of mass is given by :2

1 1 - cos - . The rate at which kinetic energy propagates 2 2

1along the string is equal to 2

m

dm dK dmv

dK dx y kx t

dK v ydt

2 2

2 2 2 2 2

avgavg

avgavg avg avg avg

cos - . The average rate

1 1cos - . As in the case of the oscillating 2 4

spring-mass syst 12

em,

m

m m

kx t

dK v y kx t v ydt

dU dK dU dKPdt dt dt dt

2 2 .mv y

(16–8)

2 2 2

2 2 2 2

1The wave equation , even though

it was derived for a transverse wave propagating alonga string under tension, is true for all types of w

y y yt t v t

The Principle of Superposition for Waves

1

2

1 1 2 2 1 2

aves.This equation is "linear," which means that if and

are solutions of the wave equation, the function is also a solution. Here and are constants.

The principle of superposition

yyc y c y c c

is a direct consequenceof the linearity of the wave equation. This principle can be expressed as follows:

1 2

Consider two waves of the same type that overlap at some point in space.Assume that the functions ( , ) and ( , ) describe the displacementsif the wave arrived at alone. The displacement at

Py x t y x t

P P

1 2( , ) ( , ) ( , ). when both waves

are present is given by Overlapping waves do not in any way alter each other's travel.

y x t y x t y x t Note : (16–10)

1 2( , ) ( , ) ( , ) y x t y x t y x t

Consider two harmonic waves of the same amplitude and frequency that propagate along the -axis. The two waves have a phase difference . Wewill combine these waves using the p

x

Interference of Waves

1

rinciple of superposition. The phenomenon of combining waves is known as and the two waves are said to . The displacement of the two waves is given by the functions: ( , )y x t y

interference,interfere

2 1 2

sin

and ( , ) sin .

, sin sin

The resulting wave has the same frequency asthe original waves, and its a

, 2 cos sin .

mplitude is

2 2

m

m

m

m

m

m

y x t y

kx t

y x t y kx t y y y

y x t y kx t y k

t

x t

kx

y

Its phase is e2 cos qual. 2

to .2my

(16–11)

The amplitude of two interfering waves is given by

2 cos . It has its maximum value if 0.2

In this case, The displacement of the resulting wave is

, 2

2 .

s

m

m

m m

m

y y

y y

y x t y

Constructive Interference

in .2

This phenomenon is known as

kx t

fully constructive interference.

(16–12)

The amplitude of two interfering waves is given by

2 cos . It has its minimum value if .2

In this case, The displacement of the result

0.ing wave is

, 0.Thi

m m

m

y y

y x

y

t

Destructive Interference

s phenomenon is known as fully destructive interference.

(16–13)

The amplitude of two interfering waves is given by

2 cos . When interference is neither fully2

constructive nor fully destructive it is called

An

m my y

Intermediate Interference

intermediate interference.

2example is given in the figure for . 3

In this case, The displacement of the resulting wave is

, sin .3

Sometimes the phase difference is expressed as a diffe ence

.

r

m

m m

y x t y kx t

y y

Note :in wavelength .

In this case, remember that 2 radians 1 .

(16–14)

Example 5:

1 2

Consider the superposition of two waves that have the samefrequency and amplitude but travel in opposite directions. The displacements of two waves are , sin and ,my x t y kx t y x t

Standing Waves :

1 2

sin .

The displacement of the resulting wave , , ,

, sin sin 2 sin cos .

This is not a traveling wave but an oscillation that has a position- dependent amplitude.

m

m m m

y kx t

y x t y x t y x t

y x t y kx t y kx t y kx t

It is known as a standing wave.

, 2 sin cosmy x t y kx t

(16–17)

n n n

a

a

a

n

The displacement of a standing wave is given by the equation

, 2 sin cos .

The position-dependent amplitude is equal to

These are defined as position

2 sin

s where the standing wave

.m

m

y x t y kx

y kx

t

Nodes. amplitude vanishes. They occur when for 0,1,2,

2 for 0,1, 2,....

These are defined as positions where the standing wave amplitude is maximum.

They occur when

2n

kx n n

x n n

kx

x n

Antinodes.

1 for 0,1, 2,....2

2 1 for 0,1, 2,....2

The distance between adjacent nodes and antinodes is /2. The distance between a node and an a

1

djac

2 2n

n n

x n nx n

Note 1 : Note 2 : ent antinode is /4.

(16–18)

Reflections at a Boundary

a) Hard reflection : A pulse incident from the right is reflected at the left end of the string, which is tied to a wall. Note that the reflected pulse is inverted from the incident pulse

b) Soft reflection :Here the left end of the string is tied to a ring that can slide without friction up and down the rod. Now the pulse is not inverted by the reflection.

Consider a string under tension that is clamped at points and separated by a distance . We senda harmonic wave traveling to the right. The wave isreflected at point

A B L

B

Standing Waves and Resonance

and the reflected wave travels to the left. The left-going wave reflects back at point and creates a thrird wave traveling to the right. Thus we have a large number of overlapping waves, half of w

A

hich travel to the right and the rest to the left.

A

A

A

B

B

B

For certain frequencies the interference produces a standing wave. Such astanding wave is said to be . The frequencies at which the standingwave occurs are known as the

at resonanceresonant frequenci of the system. es

(16–19)

A

A

A

B

B

B

Resonances occur when the resulting standing wavesatisfies the boundary condition of the problem. These are that the amplitude must be zero at point and point and arise from the fact that the strin

AB

1

1

g isclamped at both points and therefore cannot move. The first resonance is shown in fig. a. The standing

wave has two nodes at points and . Thus 2

. The second standing wave is shown2

A L

L

B

2

in fig. b. It has three nodes (two of them at and ).

In this case 2 2

.

A

L

B

L

3

The third standing wave is shown in fig. c. It has four nodes (two of them at and ).

In this case 3 The general expression for the resonant2

2 wavelengths is for

2 .

1, 2,

3

n

A B

L L

L nn

3,.... The resonant frequencies .2n

n

v vf nL

(16–20)

Example 6:

Example 7:

Example 8:

Example 9:

Example 10:

Example 11:

Example 12:

Example 13:

Example 14:

Example 15:

Example 16:

Example 17:

Example 18:

Example 19:

Example 20:

Example 21:

Example 22:

Example 23:

Example 24: