Chapter 16 Waves I

In 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)

A wave is defined as a disturbance that is self-sustained and propagates in space with a constant speed

Waves can be classified in the following three categories:

1. Mechanical waves. These involve motions that are governed by Newton’s laws and can exist only within a material medium such as air, water, rock, etc. Common examples are: sound waves, seismic waves, etc. 2. Electromagnetic waves. These waves involve propagating disturbances in the electric and magnetic field governed by Maxwell’s equations. They do not require a material medium in which to propagate but they travel through vacuum. Common examples are: radio waves of all types, visible, infra-red, and ultra-violet light, x-rays, gamma rays. All electromagnetic waves propagate in vacuum with the same speed c = 300,000 km/s

3. Matter waves. All microscopic particles such as electrons, protons, neutrons, atoms etc have a wave associated with them governed by Schroedinger’s equation. (16 – 2)

Waves can be divided into the following two categories

depending on the orientation of the disturbance with

respect to the wave propagation velocity .

If the disturba

v

Transverse and Longitudinal waves

r

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

that

transverse

propagates along a string. The movement of each

particle on the string is along the -axis; the wave itself

propagates along the -axis.

y

x

A wave in which the associated disturbance is parallel to the wave propagation

velocity is known as a " ". An example of such a wave is given

in the lower figure. It is produced by a p

lonitudinal wave

iston 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 .vr

(16 – 3)

Consider the transverse wave propagating along

the string as shown in the figure. The position of

any point on the string can be described by a

function ( , ). Further along the chapter

we shall s

y h x t=

( )

ee that function has to have a specific

form to describe a wave. Once such suitable

function is: ( , ) sin -

Suchm

h

y x t y kx tω= a wave which is described by a sine

(or a cosine) function is known as

" ".

The various terms that appear in the expression

for a harmonic waveare identified in the lower figure

Function (y x

harmonic wave

( )

, ) 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

.

o

o o

o

t x t

t t

t t y y x t

x x

== =

= ( )In this case , .oy y x t=

(16 – 4)

( )

( )

The is the absolute value of the

maximum displacement from the equilibrium

position.

The is defined as the argument

of the sine function.

The

( , ) sin

is the

m

my

kx

y x t y kx t

t

ω

ω

λ

−

−

=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 ed

2

fix

m m m

t t y x y x

y kx y k x y kx k

k

T

k

λλ λ

π λ πλ

π

= = + →

= + = +

→ = → =

period

( ) ( ) ( )

) tthe sine function to complete

one oscillation. We take 0 (0, ) (0, )

sin sin sin 22

m m m

x

x y t y t T

y t y t T y t T TT

πωω ω ω ω ω π

= → = + →

− = − + = − + → = → =

(16 – 5)

In the figure we show two snapshots of a harmonic wave

taken at times and . During the time interval

the wave has traveled a distance . The wave speed

. On

t t t t

x

xv

t

+ ∆ ∆∆

∆=∆

The speed of a traveling wave

e method of finding is to imagine that

we move with the same speed along the -axis. In this

case the wave will seem to us that it 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 harmonic

my x t y kx t

kx t t

dx dx dxk vdt dt k dt k

ωω

ω ωω

= −− =

− = → = = =

( ) ( ) 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

x

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

x

ω ω= + = − negative

( )ral wave that propagates along the

-axis is described by the equatio ( , )n: y x t hx kx tω= +negative

vk

ω=(16 – 6)

Below we will determine the speed of a wave that

propagates along a string whose linear mass density

is . The tension on the string is equal to .

Consider a small sec

µ τ

Wave speed on a stretched string

tion of the string of length . ∆l

( )The shape of the element can be approximated to be an arc of a circle of radius R

whose center is at O. The net force in the direction of O is 2 sin .

Here we assume that 1 sin 2

F

FR R

τ θ

θ θ θ τ

=∆ ∆→ = → = (l l

= ;

( )

( )

2 2

2

The force is also given by Newton's second law:

If we compare equations 1 and 2 we get:

The speed depends on the tension and the mass densit

v vF m

R R

vv

R R

v

τ

µ

µ τ

τµ

=

=

∆ ∆

∆∆ = →

eqs.1)

= (eqs.2)

Note :

l

ll

y

but not on the wave frequency .f

µ

vτµ

=

(16 – 7)

( )

( )

Consider a transverse wave propagating along a string

which is described by the equation:

( , ) sin - . The transverse velocity

- cos - At point a both

m

m

y x t y kx t

yu y kx t

t

ω

ω ω

=∂= =∂

Rate of energy transmission

and are equal to zero. At point b both and

have maxima.

y u y u

( )

2

2

2

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

21 1

- cos - The rate at which konetic energy propagates 2 2

1along the string is equal to

2

m

dm dK dmv

dK dx y kx t

dKv

dt

ω ω

µ ω

=

=

= ( )

( )

2 2

2 2 2 2 2

cos - The average rate

1 1cos - As in the case of the oscillating

2 4

spring-mass system

m

m mavgavg

avgavg avg avg a

y kx t

dKv y kx t v y

dt

dU dK dU dKP

dt dt dt dt

ω

µ ω ω µ ω = =

= → = + 2 21

2gm

v

v yµ ω=

(16 – 8)

Consider a string of mass density and tension

A transverse wave propagates along the string.

The transverse motion is described by ( , )

Consider an element of length and mas

y x t

dx

µ τThe wave equation

( )

1 2

2 2 1 1 2 1

1 2 1 11

2 2

s

The forces

the net force along the y-axis is given by the equation:

sin sin sin sin Here we

assume that 1 and 1 sin tan

and sin tan

ynet

dm dx

F F

F F F

y

x

y

x

µτ

θ θ τ θ θ

θ θ θ θ

θ θ

== =

= − = −

∂ → = ∂ ∂= ∂

= = ;

;2 2 1

ynet

y yF

x xτ

∂ ∂ → = − ∂ ∂

θ2

2

22 1

2 2 2 22 1

2 2 2 2 22 1

From Newton's second law we have:

1

ynet y

y y yF dma dx

t x x

y yy y y y y yx x

dxx x t dx t t v t

µ τ

µτ µτ

∂ ∂ ∂ = = = − → ∂ ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ − = → = = = ∂ ∂ ∂ ∂ ∂ ∂

2 2

2 2 2

1y y

t v t

∂ ∂=∂ ∂

(16 – 9)

2 2 2

2 2 2 2

1The wave equation even though

it was derived for a transverse wave propagating along

a string under tension, is true for all types of wa

y y y

t t v t

µτ

∂ ∂ ∂= =∂ ∂ ∂

The principle of superposition for waves

1

2

1 1 2 2 1 2

ves.

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

y

y

c y c y c c+ is a direct consequence

of 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 P in space.

Assume that the functions ( , ) and ( , ) describe the displacements

if the wave arrived at P alone. The displacement at P

y x t y x t

1 2

when both waves

are present is given by:

Overlapping waves do not in any way alter the travel of each oth

( , ) ( , ) ( , )

er

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 which propagate along the x-axis. The

two waves have a phase difference . We

will combine these waves using the

φ

Interference of waves

1

principle of

superposition. The phenomenon of combing waves

is knwon as and the two waves are

said to . The displacement of the two waves

are given by the functions: ( , ) my x t y=

interference

interfere

( )( )

( ) ( ) ( )

( )

2 1 2

sin

and ( , ) sin .

, sin sin

The resulting wave has the same frequency as

the original waves, and i

, 2 cos sin

ts amplitude

2 2

2 c

m

m m

m

m

m

y x t y kx t

kx t

y x t y kx t y y y

y x t y kx t y t

y y

kx

ωω φ

ω ω φφ φω

−′= − + = +

′ = −

′ = − +

+

=

−

′

+

Its phase is equalos 2

to 2

φ φ

(16 – 11)

( ) [ ]

The amplitude of two interefering waves is given by:

2 cos It has its maximum value if 02

In this case

The displacement of the resul

2

ting wave is:

, 2 s

m m

m m

m

y y

y y

y x t y

φ φ

′ =

′ = =

′ =

Constructive interference

in2

This phenomenon is known as

kx tφω − +

fully constructive interference

(16 – 12)

( )

The amplitude of two interefering waves is given by:

2 cos It has its minimum value if 2

In this case

The displacement of the resulting wave is:

, 0

0

This

m

m my y

y x t

y

φ φ π

=

′ = =

′ =

′

Destructive interference

phenomenon is known as

fully destructive interference

(16 – 13)

The amplitude of two interefering waves is given by:

2 cos When interference is neither fully2

constructive nor fully destructive it is called

An

m my yφ′ =

Intermediate interference

intermediate interference

( ) [ ]

2 example is given in the figure for

3In this case

The displacement of the resulting wave is:

, sin3

Sometimes the phase difference is

expressed as a difference i

m

m m

y x t y kx t

y y

πφ

πω

=

′ = − +

′

=

Note :

n wavelength

In this case re

2

membre that:

radians 1π λ

λ

↔

(16 – 14)

( ) ( )1 1

A phasor is a method for representing a wave whose

diasplacement is: , sin

The phasor is defined as a vector with the following

properties:

Its magnitude is equal to the wave's am

my x t y kx tω= −

Phasors

1. 1plitude

The phasor has its tail at the origin O and rotates in

the clockwise direction about an axis through O

with angular speed .

Thus defined, the projection of the phasor on the -axis

(i.e.

my

y

ω

2.

( )

( ) ( )

1

2 2

its y-component) is equal to sin

A phasor diagram can be used to represent more than one

waves. (see fig.b). The displacement of the second wave is:

, sin The phasor of the

m

m

y kx t

y x t y kx t

ω

ω φ

−

= − + second

wave forms an angle with the phasor of the first wave

indicating that it lags behind wave 1 by a phase angle .

φφ

(16 – 15)

( )1 1

Consider two waves that have the same frequency

but different amplitudes. They also have a phase

difference . The displacements of the two waves

are: , sinmy x t y kx

φω= −

Wave addition using phasors

( )( ) ( )

( )

2 2

and

, sin . The superposition of the two

waves yields a wave that has the sameangular frequency

and is described by: sin Here is the

wave amplitude and is the phas

m

m m

t

y x t y kx t

y y kx t y

ω φω

ω ββ

= − +

′ ′ ′= − +e angle.

To determine and

we add the two phasors representing the waves as vectors

(see fig.c).

Note: The phasor mathod can be used to add vectors that

have different amplitudes.

my β′

(16 – 16)

( ) ( ) ( )1 2

Consider the superposition of two waves that have the same

frequency and amplitude but travel in opposite directions. The displacements

of two waves are: , sin , ,my x t y kx t y x t yω= − =

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. It i

m

m m m

kx t

y x t y x t y x t

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

ω

ω ω ω

+′ = +

′ = − + + =

s known as a standing wave.

( ) [ ], 2 sin cosmy x t y kx tω′ =

(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 dependant amplitude is equal to

These are defined as positions

2 si

where the stand

:

ing

wa

n

ve a

m

my kx

y x t y kx tω′ =

Nodes :

mplitude vanishes. They occur when 0,1, 2,

2 0,1, 2,...

These are defined as positions where the standing

wave amplitude is maximum.

1They occur when

2

2

n

kx n n

x n n nx

kx n

λπ

π πλ

= =

→ = → =

= +

=

Antinodes :

0,1,2,...

2 1 0,1, 2,...

2

The distance between ajacent nodes and antinodes is /2

The

1

2 2

distance between a node and an ajacent antinode is /4

n

n

x n x nn

π

π π

λ

λλ

λ

= → = + ′ = +→ =

Note 1 :

Note 2 :

(16 – 18)

Consider a string under tension on which is clamped

at points A and B separated by a sistance . We send

a harmonic wave traveling the thr right. the wave is

reflected at po

L

Standing waves and resonance

int B and the reflected wave travels to

the left. The left going wave reflects back at point A

and creates a thrird wave traveling to the right. Thus

we have a large number of overlapping waves half

of which 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 a

standing wave is said to be . The frequencis at which the standing

wave occurs are known as the

at resonance

resonant frequencie of the system. s

(16 – 19)

A

A

A

B

B

B

Resonances occur when the resulting standing wave

satisfies the boundary condition of the problem.

These are that the Amplitude must be zero at point A

and point B and arise from the fact that the strin

1

1

g is

clamped at both points and therefore cannot move.

The first resonance is shown in fig.a. The standing

wave has two nodes at points A and B. Thus 2

. The second standing wave is sh2 own

L

L

λ

λ =

=

→

2

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

In this case 2 2

L Lλ λλ = = → =

3

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

In this case 3 The general expression for the resonant2

2 wavelen

2

gths is: 1, 2,3,

3

..

n

LL

Ln

n

λ

λ

λλ = =

=

=→

= . the resonant frequencies 2n

n

v vf n

Lλ= =

(16 – 20)