Mechanical waves

Dr. Eng. Hesham SalamahDr. Eng. Hesham Salamah

hesham_2000eg@yahoo.comhesham_2000eg@yahoo.com

Please, Switch off the Please, Switch off the MobileMobile

R

emem

ber

to

Rem

embe

r to

ta

ke n

otes

take

not

es

Waves Waves

Wave is everywhere in nature!

Why can you hear me?

Why can you receive a TV signal emitted

from a TV station?

• A wave is the motion of a disturbanceA wave is the motion of a disturbance

Types of WavesTypes of Waves

Waves are classified into different

types according to their natures :

Transverse waves Longitudinal waves

Mechanical waves

Transverse waves

Electromagnetic waves

WAVES

Types of Waves; continueTypes of Waves; continue

Classification According to natures of matter:

1. 1. Mechanical Waves Mechanical Waves

Examples: water waves, sound waves, and seismic waves.

They are governed by Newton’s laws, and they can exist

only within a material medium, such as water, air, and

rock.

These waves have certain central features:

2. 2. Electromagnetic WavesElectromagnetic Waves

Examples: Visible and ultraviolet light, radio and

television waves, microwaves, X rays, and radar

waves.

They don’t require material medium to exist. These

waves have a limited wave speed in vacuum,. m/s 458 792299c

3. 3. Matter WavesMatter Waves

Examples: Electrons, protons, and other fundamental

particles; ultracold atoms.

The most remarkable property of the matter waves is

that wave functions of matter waves are referred to

as probability amplitudes of waves.

Classification According to Oscillation Types:

1. 1. Transverse WavesTransverse Waves

The direction of oscillation of medium elements is

perpendicular to the direction of travel of the wave.

2. 2. Longitudinal WavesLongitudinal Waves

The direction of oscillation of medium elements is

parallel to the direction of the wave’s travel, the motion

is said to be longitudinal, and the wave is said to be a

longitudinal wave.

Both a transverse wave and a longitudinal wave are said tobe traveling waves because they both travel from one pointto another, as from one end of the string to the other end.

Creation of Waves and PropagationCreation of Waves and Propagation

For a source of simple harmonic oscillation with angular frequency , the traveling in the positive direction of x axis and oscillating parallel to the y axis, the displacementy of the element located at position x at time t is given by

Conditions of a Mechanical waveConditions of a Mechanical wave

1. A Wave Source.

2. Medium in which a mechanical wave propagates.

A wave means that the oscillation state propagates with time. It does not mean the particles of medium move forward with wave. Particles of the medium oscillate only around their corresponding equilibrium positions.

).sin(),( tkxytxy m

Displacement Phase

Angularfrequency

Angular wave number

Amplitude ym is the amplitude ofa wave.

The phase of the waveis the argument

kx – t)

λ

2π k

f π2 T

2π ω Move to right

Space Periodicity and Time PeriodicitySpace Periodicity and Time Periodicity

Wavelength and Angular Wave Number

The wavelength of a wave is the distance (parallel to

the direction of the wave’s travel) between repetitions of

the shape of the wave (or wave shape).

According to the property of sine function, it begins to

repeat itself when its angle is increased by 2 rad,

That is

2

k (Angular wave number)

Period, Frequency, and Angular Frequency

You can choose any time t1, through a period, the

oscillation is repeated.

The period of oscillation T of a wave is defined to be the

time any mass element takes to moves through one full

oscillation.

).sin(sin 11 Ttyty mm

This can be true only ifT =2or

.2T

Angular frequency

Frequency ƒ is defined by

.2

1

Tf

Wavefronts

Wavefronts are surfaces over which the oscillations of wave have

the same value; usually such surfaces are represented by whole

or partial circles in a two-dimensional drawing for a point

source.

If point A retains its displacement as it moves, its phase must remain a constant, that is:

constant. a kxt

Taking a derivative, we have

,0dtdx

k

or.

kv

dtdx

By using definitions of and k, we can rewrite v as

fT

v

Phase speed

The Speed of a Traveling WaveThe Speed of a Traveling Wave

Question:

How to describe a wave traveling in negative direction of x?

Similarly to the discussion of positive direction of x, the

wave speed of a wave traveling in the negative x direction

should be have a form:

kdtdx

v

This corresponding to the condition:

constant a kxt

This means that the wave function should have a form:

)sin(),( tkxytxy m

It seems that k may be have a property of vector, its

magnitude is

vk

and its direction is in that of wave traveling, or wave

velocity.

For a traveling wave to right, this means the oscillation at x is:

For a traveling wave to left, this means the oscillation at x is:

)sin(),( tkxytxy m

)sin(),( tkxytxy m

(a) (b)

A BF F

v

Wave Speed on a Stretched StringWave Speed on a Stretched String

v

The speed of a wave along a stretched ideal string depends only on the tension and linear density of the string and not on the frequency of the wave.

DensityMassLinear

ForceAppliedv

L

m

Length

mass

Energy and Power Energy and Power of a Traveling String Waveof a Traveling String Wave

In general, the average energy flow through an unit area of

perpendicular to the propagating direction, or density of energy

flow, or intensity of the wave, is given by:

22 2

1 maverage yvP

Where is the linear mass density of wave.

v

Example (1): Example (1): The equation of a wave is

m

4

)4010(2

sin05.0),( txtxy

Find: (a) the wavelength, the frequency, and the wave velocity;

(b) the particle velocity and acceleration at x = 0.5 m and t = 0.05 s.

k = 2/ = 5 rad/m and so = 0.4 m.

The angular frequency is = 2 f = 20 rad/s; thus f = 10 Hz.

The wav velocity is υ = f = /k = 4 m/s in the x direction.

(b) The particle velocity and acceleration are

m/s22.2)42

5(cos)05.0()20(

t

y

222

2

m/s140)42

5(sin)05.0()20(

t

ya

Example (2): Example (2): One end of a string is fixed. It hangs over a

pulley and has a block of mass 2.00 kg attached to the other

end, as in Fig. 2.4. The horizontal part has a length of 1.6 m

and a mass of 20.0 g. What is the speed of a transverse pulse

on the string?

m

L = 1.6 m

Solution:Solution:

The tension is simple the weight of the block; that is, F = 19.6 N.

The linear mass density is (20 103)/(1.6) = 1.25 102 kg/m.

m/skg/m

N9.66.39

1025.1

12

F

v

Example (3): Example (3): A rod vibrating at 12 Hz generates harmonic waves with

an amplitude of 1.5 mm in a string of linear mass density 2 g/m. If the

tension in the string is 15 N, what is he average power supplied by the

source?

m/skg10(2

N3-

6.8615

F

v

The angular frequency is rad/s.4.752 f

mW1.1)6.86()105.1()4.75()102(2

1 2323

2

2

1av

vyP m

Interference of

waves

Dr. Eng. Hesham SalamahDr. Eng. Hesham Salamah

hesham_2000eg@yahoo.comhesham_2000eg@yahoo.com

Please, Switch off the Please, Switch off the MobileMobile

Suppose that two waves travel simultaneously along the same stretched string. Let y1(x, t) and y2(x, t) be the displacements that the string would experience if each wave traveled alone. The displacement of the string when the waves overlap is then the algebraic sum:

).,(),(),( 21 txytxytxY

It means

Overlapping waves algebraically add to produce a resultant wave (or net wave).

Superposition for WavesSuperposition for Waves

The Principle of Superposition for Waves

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

Interference of Waves

Suppose that there are two sinusoidal

waves of the same wavelength and

amplitude in the same direction along

a stretched string.

)sin(),(1 tkxytxy m

)sin(),(2 tkxytxy m

Where, is the phase shift

)2

1sin(]

2

1cos2[

)sin()sin(

),(),(),( 21

tkxy

tkxytkxy

txytxytxY

m

mm

The new resultant amplitude resultant amplitude depends on phase :

1. If = 0 rad, the two interfering waves are exactly in phase.

2

1cos2 mm yY

)sin(2),( tkxytxY m

In this case, the interference produces the greatestpossible amplitude which is called fully constructiveinterference.

From the principle of superposition, the resultant wave is:

2. If = 0),( txY

Although we send two waves along the string, we see no

motion of the string. This type of interference is called

fully destructive interference (see fig. (e)).

3. When interference is neither fully constructive nor

fully destructive, it is called intermediate interference.

Figures (c) and (f) for =2/3.

Two waves with the same wavelength are in phase if

their phase difference is zero or any integer number of

wavelengths. Thus, the integer part of any phase

difference expressed in wavelengths may be discarded.

According to the previous section, the condition for

fully constructive interference is

.210for ),2( ,,,nn

Similarly, fully destructive interference occurs when

.,2,1,0for ,)12( nn

Standing WavesStanding Waves

Let’s consider two waves with the same oscillating

frequencies, but traveling in opposite directions.

To analyze a standing wave, we represent the two

waves with the equations:

The principle of superposition gives, for the combinedwaves,

tkxy

tkxytkxytxY

m

mm

cos]sin2[

)sin()sin(),(

)sin(),( tkxytxy m

)sin(),( tkxytxy m

The wave like this — two sinusoidal waves of the same amplitude and wavelength travel in opposite directions, their interference with each other produces a standing wave.

Standing Waves and Resonance Standing Waves and Resonance

• For For certain frequenciescertain frequencies, , the interference produces a the interference produces a standing wave pattern (or standing wave pattern (or an an oscillation mode)oscillation mode) Such a standing wave is Such a standing wave is said to be produced at said to be produced at resonanceresonance, and the string , and the string is said to is said to resonateresonate at these at these certain frequencies, called certain frequencies, called resonant frequenciesresonant frequencies. .

For n=1, the frequency f =v/2L. This Lowest frequency

is called the fundamental mode or the first harmonic .

For n=2, we have the second harmonic oscillation mode.

For n=3, we have the third harmonic oscillation mode.

)2

t ω sin(k x 2

cos y 2 t)y(x, m

4

t 40 x 3sin 0.17 4

t 40 x 3sin 4

cos 0.12 * 2 t)y(x,

s 0.05 T T

π2 40

T

π 2 ω

m/s 13.33 3

40

k

ω υ

Example (1):Example (1):

The following two waves interfere :

y1 = 0.12 sin[3x – 40t], y2 = 0.12 sin[3x – 40t – 0.5)

•Find the equation of the resultant wave

•What is the periodic time of the wave

•What is the speed of the resultant wave

)0.20.3(sin0.4(1 txy cm)

2y ),0.20.3(sin0.4( tx cm)

Example (2): Example (2): Two waves traveling in opposite directions produce a standing wave. The individual wave functions are

and

where x and y are measured in cm. Find: (a) the amplitude of the simple harmonic motion

of the element of the medium located at x = 2.3 cm; (c) the maximum displacement of an

element located at an antinode?

Here, ym = 4.0 cm, k = 3.0 rad/cm and = 2.0 rad/s.

) cos(2.0 ) sin(3.0 )0.8()(sin()cos(2),( txkxtytxy m

cm63.4]180

(2.3) [(3.0)sin cm) 0.8( cm3.2 xamplitudeThe

(b) The maximum displacement of an element at an antinode is the amplitude of

the standing wave. From part (a) of the recent example, we have the value of this

maximum being 8.0 cm.

Hz 2 12

24 1 f

Hz 4 6

24 2 f

Hz 6 4

24 3 f

Example (3)Example (3)

The speed of a wave on a particular string is 24 m/s. if the

string is 6 m long, what is the fundamental frequency, first

harmonic and second harmonic?