Chapter 1

Wave Motion

1

Dr Mohamed Saudy

Why learn about waves?

Waves carry useful

information and energy.

Waves are all around

us:

light from the stoplight

electricity flowing in wires

radio and television and

cell phone transmissions 2

Definition: A wave is a traveling disturbance in some physical system.

Wave Motion : Basic Concept

Alternatively, a periodic disturbance that travels through space and time

3

Wavelength visualized Parameters of a Wave

–Wavelength l (e.g., meters)

–Frequency f (cycles per sec Hertz)

–Propagation speed c (e.g., meters / sec)

Characteristics:

C= f l

4

Wave Motion

5

Wave Motion : Classification

Traveling waves – Disturbance

moves along the direction of wave propagation

Standing waves - Disturbance

oscillates about a fixed point.

WAVE MOTION

Standing/Stationary Wave

E. M. WAVES

Longitudinal Wave

Transverse Wave

Mechanical Waves

Traveling Wave

Waves can be characterized as

Transverse or Longitudinal.

6

Types of Waves

There are two main types of waves

Mechanical waves

Some physical medium is being disturbed

The wave is the propagation of a disturbance through

a medium (such as water, air and rock)

Examples: water waves, and sound waves

Electromagnetic waves (E.M Waves)

No medium required

Examples are light, radio waves, x-rays

All e.m waves travel through the vacuum at the same

speed. 7

General Features of Waves

In wave motion, energy is transferred over a

distance

Matter is not transferred over a distance

All waves carry energy

The amount of energy and the mechanism

responsible for the transport of the energy differ

8

Mechanical Wave

Requirements

Some source of disturbance

A medium that can be disturbed

Some physical mechanism through which

elements of the medium can influence each

other

9

Transverse Wave

A traveling wave or pulse

that causes the elements of

the disturbed medium to

move perpendicular to the

direction of propagation is

called a transverse wave

The particle motion is

shown by the blue arrow

The direction of propagation

is shown by the red arrow

10

Longitudinal Wave

A traveling wave or pulse that causes the elements of the disturbed medium to move parallel to the direction of propagation is called a longitudinal wave

The displacement of the coils is parallel to the propagation

11

Complex Waves

Some waves exhibit a combination of transverse

and longitudinal waves

Surface water waves are an example

Use the active figure to observe the displacements

12

Example: Earthquake Waves

(Complex wave)

P waves “P” stands for primary

Fastest, at 7 – 8 km / s

Longitudinal

S waves “S” stands for secondary

Slower, at 4 – 5 km/s

Transverse

A seismograph records the waves and allows determination of information about the earthquake’s place of origin

13

• An ocean wave is a

combination of transverse and

longitudinal.

• The individual particles move

in ellipses as the wave

disturbance moves toward the

shore.

Water Waves

14

Sinusoidal Waves

The wave represented by

the curve shown is a

sinusoidal wave

It is the same curve as sin q

plotted against q

This is the simplest

example of a periodic

continuous wave

It can be used to build more

complex waves

15

Terminology: Amplitude and

Wavelength

The crest of the wave is the location of the maximum displacement of the element from its normal position This distance is called

the amplitude, A

The wavelength, l, is the distance from one crest to the next

16

Terminology: Wavelength and

Period

More generally, the wavelength is the

minimum distance between any two identical

points on adjacent waves

The period, T , is the time interval required for

two identical points of adjacent waves to pass

by a point

17

Terminology: Frequency

The frequency, ƒ, is the number of crests (or

any point on the wave) that pass a given

point in a unit time interval

The time interval is most commonly the second

18

Terminology: Frequency, cont

The frequency and the period are related

When the time interval is the second, the

units of frequency are s-1 = Hz

Hz is a hertz

1ƒ

T

19

period ( T ) units - time

frequency ( f ) units - 1/time

time per wave

waves per time

T = 1

f f =

1

T v = = f

T

l l

if f = 10

sec , then T =

10 sec

1

sec

1 =

sec

cycle = hz

Wave Motion : Properties

20

Terminology, Example

The wavelength, l, is

40.0 cm

The amplitude, A, is

15.0 cm

The wave function can

be written in the form

y = A cos(kx – t)

21

Wave Equations

We can also define the angular wave number

(or just wave number), k

The angular frequency can also be defined

2k

l

22 ƒ

T

22

Wave Equations, cont

The wave function can be expressed as

y = A sin (k x – t).

The speed of the wave becomes v = l ƒ.

If x 0 at t = 0, the wave function can be

generalized to

y = A sin (k x – t + )

where is called the phase constant.

23

Speed of a Wave on a String

The speed of the wave depends on the physical characteristics of the string and the tension to which the string is subjected

This assumes that the tension is not affected by the pulse

This does not assume any particular shape for the pulse

tension

mass/length

Tv

24

Example 1: A wave pulse on a string moves a distance of 10 m

in 0.05 s.

(a) What is the velocity of the pulse?

(b) What is the frequency of a periodic wave on the same

string if its wavelength l is 0.8 m?

Solution: (a) The velocity of the pulse is C=x/t, where

x= 10 m, t=0.05 s, C= 200 m/s

(b) The periodic wave has the same velocity 200 m/s,

f=C/l=250 Hz= 250 s-1

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Example 2: The tension on the longest string of

a grand piano is 1098 N, and the mass per unit

length is 0.065 Kg/m. What is the velocity of a

wave on this string?

Solution

26

Example 3: An electromagnetic vibrator sends waves down a

string. The vibrator makes 600 complete cycles in 5 s. For one

complete vibration, the wave moves a distance of 20 cm. What

are the frequency, wavelength, and velocity of the wave?

Solution

The distance moved during a time of one cycle is the

wavelength; therefore: l=0.2 m

The velocity of wave

27

120 0.2 24 /v f Hz m m sl

Energy in Waves in a String

Waves transport energy when they propagate

through a medium

We can model each element of a string as a

simple harmonic oscillator

The oscillation will be in the y-direction

Every element has the same total energy

28

Energy, final

the total kinetic energy in one wavelength is

Kl = ¼2A 2l

The total potential energy in one wavelength

is Ul = ¼2A 2l

This gives a total energy of

El = Kl + Ul = ½2A 2l

29

Power Associated with a Wave

The power is the rate at which the energy is being transferred:

The power transfer by a sinusoidal wave on a string is proportional to the Frequency squared

Square of the amplitude

Wave speed

l

2 2

2 2

1122

AEnergy E

A vTime t T

30

Example 4: A 2 m string has a mass of 300 g and

vibrates with a frequency of 20 Hz and an amplitude

of 50 mm. If the tension in the rope is 48 N, how

much power must be delivered to the string?

Solution

31

The Superposition Principle • When two or more waves (blue and green) exist in

the same medium, each wave moves as though the other were absent.

• The resultant displacement of these waves at any point is the algebraic sum (yellow) wave of the two displacements.

Constructive Interference Destructive Interference

Formation of a

Standing

Wave:

Incident and reflected

waves traveling in

opposite directions

produce nodes N and

antinodes A.

The distance between

alternate nodes or anti-nodes

is one wavelength.

Possible Wavelengths for Standing

Waves

Fundamental, n = 1

1st overtone, n = 2

2nd overtone, n = 3

3rd overtone, n = 4

n = harmonics

2 1, 2, 3, . . .n

Ln

nl

Possible Frequencies f = v/l :

Fundamental, n = 1

1st overtone, n = 2

2nd overtone, n = 3

3rd overtone, n = 4

n = harmonics

f = 1/2L

f = 2/2L

f = 3/2L

f = 4/2L

f = n/2L

1, 2, 3, . . .2

n

nvf n

L

Characteristic Frequencies

Now, for a string under

tension, we have:

and 2

F FL nvv f

m L

Characteristic frequencies:

; 1, 2, 3, . . .2

n

nf n

L

37

Example 5: A uniform cord has a mass of 0.3 kg and a length of 6

m. The cord passes over a pulley and supports a 2 kg object. Find

the speed of a pulse traveling along this cord?

Solution: The tension in the cord is equal to the weight of the

suspended M=2 kg object:

=Mg= (2 Kg)(9.8 m/s2)=19.6 N

The mass per unit length of the cord is

Therefore, the wave speed is

38

Example 6: A 9-g steel wire is 2 m long and is under a tension of

400 N. If the string vibrates in three loops, what is the

frequency of the wave?

Solution: For three loops: n = 3

Summary for Wave Motion:

Lv

m

1f

T

; 1, 2, 3, . . .2

n

nf n

L

2 21, 2

2E A f l

v f l

Multiple Choice

40

1. The wavelength of light visible to the human eye is on the

order of 5 10–7 m. If the speed of light in air is 3 108 m/s, find

the frequency of the light wave.

a. 3 107 Hz

b. 4 109 Hz

c. 5 1011 Hz

d. 6 1014 Hz

e. 4 1015 Hz

2. The speed of a 10-kHz sound wave in seawater is

approximately 1500 m/s. What is its wavelength in sea water?

a. 5.0 cm

b. 10 cm

c. 15 cm

d. 20 cm

e. 29 cm

41

3. If y = 0.02 sin (30x – 400t) (SI units), the frequency of the wave is

a. 30 Hz

b. 15/ Hz

c. 200/ Hz

d. 400 Hz

e. 800 Hz

4. If y = 0.02 sin (30x – 400t) (SI units), the wavelength of the wave

is

a. /15 m

b. 15/ m

c. 60 m

d. 4.2 m

e. 30 m

5. If y = 0.02 sin (30x – 400t) (SI units), the velocity of the wave is

a. 3/40 m/s

b. 40/3 m/s

c. 60/400 m/s

d. 400/60 m/s

e. 400 m/s

42

6. A piano string of density 0.005 kg/m is under a tension of 1350 N. Find

the velocity with which a wave travels on the string.

a. 260 m/s

b. 520 m/s

c. 1040 m/s

d. 2080 m/s

e. 4160 m/s

7. A 100-m long transmission cable is suspended between two towers. If

the mass density is 2.01 kg/m and the tension in the cable is 3 104 N,

what is the speed of transverse waves on the cable?

a. 60 m/s

b. 122 m/s

c. 244 m/s

d. 310 m/s

e. 1500 m/s