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WAVE MOTION-1

Definition

- A wave is a periodic disturbance which propagate energy from a point in to another.
- Waves transfer energy from point to another point without carrying matter.

CLASSIFICATION OF WAVES

Waves may be classified into two classes:

- Mechanical waves
- Electromagnetic waves

MECHANICAL WAVES

Definition

A mechanical wave is a disturbance which is transported through a medium due to particle interaction.

or

Is the wave which require material medium for transfer of energy from one point to another.

Example

- waves on a spring
- water waves
- sound waves
- waves on stretched string (e.g. in musical instruments)

- mechanical waves require a material

medium to transfer energy. - when a mechanical waves travel through a medium the particle that make up the medium are disturb from their rest or equilibrium positions.

ELECTROMAGNETIC WAVES

Definition

Electromagnetic waves are the waves that consists of particles moving in electric and magnetic field. or

Electromagnetic waves are the waves which does not require the material medium during the propagation of energy from one point to another.

-The electric and magnetic field oscillates at right angle to each other and to the direction of propagation

DIAGRAM OF ELECTROMAGNETIC WAVES

- Electromagnetic waves do not require a material medium to transfer energy.
- They can travel through a vacuum.

Example:

- visible light
- radio waves
- infra-red radiation iv. ultraviolet radiation

- gamma radiation
- X-rays

WAVES FORM

-A wave form is a shape of a wave or pattern representing a vibration.

-It can be illustrated by drawing a graph of the periodically varying quantity against

distance for one

distance for one

complete | wavelength. |

TYPES OF Waves may be divided into two forms , which are:- | WAVE |

- Stationary / Standing waves
- Progressive waves

- STATIONARY / STANDING WAVE

Are the types of wave in which the wave profile is not moving

- PROGRESSIVE WAVE

Are the type of waves in which the wave profile is moving

TYPES OF PROGRESSIVE WAVE

i) Transverse waves ii) Longitudinal waves iii) Mechanical wave iv) Electromagnetic wave

TRANSVERSE WAVE

Definition

A transverse wave is the one that make the particle of the medium to vibrate in a direction perpendicular to the direction of movement of the wave

Example

- Water waves
- Wave on a string
- Electromagnetic waves

– A transverse wave is propagated by a series of crest C and trough (valley).

CREST

-Is an elevation of the medium above its equilibrium state when a transverse wave passes through the medium.

TROUGH (VALLEY)

-Is the depression of the medium below its equilibrium state when transverse wave passes through the medium.

LONGITUDINAL WAVES

Definition:

A longitudinal wave is the one which the particles of the material medium vibrate in a direction parallel to the direction of the wave motion.

DIAGRAM OF LONGITUDINAL WAVES

Example

- Sound waves

TRANSVERSE WAVES | LONGITUDINAL WAVES |

1 )The particle of the medium vibrate perpendicular to the direction in which the waves advance | 1) The particle of the medium vibrate in direction in which the wave advance. |

- A longitudinal wave is propagated by a series of compression and rarefaction

Where C = Compression

R = Rarefaction

COMPRESSION

- Is a region of high pressure in a longitudinal wave.

RAREFACTION

- Is a region of low pressure in longitudinal wave.

DIFFERENCE BETWEEN TRANSVERSE AND LONGITUDINAL WAVES

GENERAL WAVE DIAGRAM

- The crest and trough (valleys) in transverse wave can be likened to compression and rarefaction respectively in a longitudinal wave.
- Thus we may represent any wave by a single diagram as shown bellow:-

2 ) It consists of crest and troughs. | 2) it consists of compression and rarefaction |

3) It can propagate only in solid and at the surface of liquids | 3) it can propagate in all type of media (solid gas) |

4 ) There is no pressure variation | 4) The pressure and density are maximum at comp minimum at rarefactions. |

Where y = wa

ve displacement

ve displacement

X = direction of waves travel

T = Time of propagation

0 = speed of wave

TERMS APPLIED IN WAVE

The chief characteristic of a wave are:-

- Amplitude
- Wavelength iii. Frequency iv. Speed/ Velocity of propagation

v. v. Displacement

AMPLITUDE

- Symbol, A

-This is maximum displacement of the disturb particle from its equilibrium position.

- The SI unit of amplitude is the meter( m)

WAVE LENGTH

- Symbol, (Greek letter ―lambda‖),λ
- This is the distance between two successive points of equal phase in a wave.

Alternative definition

- The wave length of a wave is the distance between two successive or adjacent crest or trough.
- It is a distance that the wave travels in the complete cycle.

UNIT OF LAMBDA, λ

- The S.I unit of wavelength is the meter,m
- Other unit in common use are:

i) nanometer (nm) ii) angstrom (Å)

NOTE

1 nm = 10-9m

1Å = 10-10m

FREQUENCY

-Symbol, ƒ

-This is the number of complete disturbance (cycles) passing a given point in unit time.

-ƒ

UNIT OF , ƒ

- By definition

f

= Cycles/sec

= Hertz (HZ)

- Hence the S.I unit of frequency is the Hertz (HZ).

1HZ = 1 cycle/ sec Other unit in common use are:

i. Kilohertz (KHZ) ii. Megahertz (MHZ)

NOTE:

SPEED OF PROPAGATION

-Symbol, V

- This is the distance covered by the wave in unit time.

V = Distance covered

Time

V

UNIT OF V

Definition

V

-Hence the S.I unit used is meter/sec (ms-1)

PERIOD

- Symbol, T

-This is the time taken for the wave to make one complete cycle (oscillation)

- It is the time taken for the wave to travel through a distance equal one wavelength -The S.I unit for period is the second (S)

RELATIONSHIP BETWEEN ƒ AND T

-From the definition of ƒ

-This given an alternative unit of ƒ i.e. per second (S-1)

RELATIONSHIP BETWEEN V and ƒ

- From the definition of V

V = X

T

For | complete | one | cycle | ||||

x | =λ | then | t | = | T | (=period) | |

v | = | λ/T | |||||

But | f | = | 1/T | ||||

where | v | = | λf | ||||

= | velocity | ||||||

λ | = | wavelength | |||||

T = Period |

PHASE OF WAVES MOTION

- This is an angular displacement of a given wave particle in time t.
- This term is usefully in comparing two wave motions.
- Two waves are said to be in phase if their maximum and minimum values occur at the same instant otherwise there is said to be a phase difference

Two wave in phase;

PHASE DIFFERENCE / PHASE ANGLE

- Symbol, Ø
- This is the difference in phase between two wave motion
- If the phase difference is radians then the two waves are in ant phase (out of phase)

##### EXPRESSION OF PHASE DIFFERENCE

- Consider two particle O and P along the path of the waves

- Let O be origin of the wave
- A particle at P a distance X right of O will vibrate in different phase with the particle at O. – Let Ø be phase different between O and P
- Now

2 rad

##### PROGRESSIVE/ TRAVELING WAVE

-A progressive wave / traveling waves are that wave in the wave profile move along with the speed of wave profile across the medium

– Both transverse and longitudinal waves are progressive wave i.e. the wave profile move along with the speed of the wave across the medium.

-The oscillation of any progression wave repeat at equal interval of distance called wavelength and at equal interval of time called period.

-The vibration proceed to progressive waves are of the same amplitude and frequency.

##### THE PROGRESSIVE/ TRAVELING WAVE EQUATION

- Consider a progressive / traveling wave which is moving from left to right with a velocity V.

- Let O be origin of the wave

-The displacement y of the particles at O

- If the wave generate travel from left to right then a particle at P, distance X from O will lag behind the particles at O by a phase angle Ø given by

Ø. …………………………..(i)

- For the displacement y of the particle at P we have:

Consider the particle vibrating with Amplitude A

Y | is | a | displacement | t | time | t | |||

Sin | θ | = | y/a | ||||||

y | = | a | sin | θ | |||||

But | θ | = | wt | ||||||

y | = | A | sin | wt |

This is displacement at time t when initial displacement is zero. But if initial displacement is y’ =

A sin Φ

Displacement y = A sin wt + A sin Φ

y = A sin (wt + Φ)

Substitute equation (i) in this equation

Where

A = Amplitude w = Angular velocity t = time

Φ = phase difference

If the wave is traveling in the opposite direction i.e. in the negative X – direction then the equation becomes

If the wave is traveling in the same direction i.e in the positive x- direction the equation becomes

where k = 2π/ λ ƒ = frequency

T = period

Problem 01

- What is a wave?
- Y = A sin is the equation

Of a plane progressive wave where x and y are in cm and t is time in second.

Determine:

- Frequency
- Wavelength
- Velocity

Problem 02

A certain travelling wave has equation

Y = A sin (t + kx)

- Deduce whether the wave is traveling in the positive or negative x-direction.
- If = 6.6 x 103 rad s-1 and k = 20m-1 calculate the speed of the wave in the medium

##### PATH DIFFERENCE

- This is difference in path length between two waves which meet at a point or between two points in a given wave.
- It is the distance corresponding to phase difference

##### CASE 1

– Consider waves from two sources S1 and S2 which meet at point P.

Path difference, âˆ†x = S2 P- s1 p

##### CASE 2

- Consider two point P1 and P2 in the path of the wave from source

S

Path difference, âˆ†x

- from the expression of phase difference

Ø = 2πx

λ

- when x = âˆ†x = path difference, Ø

âˆ†Ø =2π

âˆ†x

âˆ†x

λ

Problem 03

A train of plane wave sound propagating in air has individual particles of air executing simple harmonic motion such as that their displacement y from equilibrium position at any time t is given by Y = 5×10-6 sin (800πt + Ø)

Where y is in cm t is in second and Ø is a phase term. Calculate a ) the wavelength of the wave

Given that velocity of sound air = 340ms-1

Problem 04

The speed of wave in a medium is 960ms-1. If 3600 waves are passing through a point in one minute, calculate the wavelength.

Problem 05

A progressive wave of frequency 500HZ is traveling with the velocity of 360ms-1. How far apart are two point 600out of phase?

Problem 06

A plane progressive wave of frequency 25HZ, amplitude 2.5 x 10-5 and initial phase zero propagate along the negative x- direction with a velocity of 300ms-1.

At any instant what is the phase difference between oscillations at two points 6 cm apart along the line of propagation?

Problem 07

The equation y = A sin ( represent a plane wave traveling in a medium along the x direction y being the displacement at the point x at time t.

- Deduce whether the wave is traveling in the positive x- direction or in the negative x- direction
- ) If a = 1.0 x 10-7m, x 103s-1 and k = 20m-1 calculate

i) The speed of the wave ii) The maximum speed of a particle of the medium due to the wave

##### SUPERPOSITION OF WAVES

- The principle state “If two or more wave arrive at a point simultaneously then the resultant displacement at that point is the vector sum of the displacement due to individual waves”.

- Where y = resultant displacement

are the displacement due to individually waves.

Problem 08

Two waves travelling together along the same line are given by:

Y1 = 5 sin ( ) and

Y2 = 5 sin (

Find:

i) The resultant equation of motion ii) The resultant amplitude

iii) The initial phase | angle | of | the motion |

IMPORTANT CASE – There are three important cases: | OF | SUPERPOSITION |

- stationary waves

- beats
- interference

##### STATIONARY WAVES/STANDING WAVES

Definition

A stationary / standing wave is a form of wave in which the profile of the wave does not move through the medium but remains stationary.

##### HOW A STATIONARY WAVE IS FORMED

-A stationary wave result when a progressive / travelling wave is reflect back along its own path.

-The incident and reflect wave then interfere to produce a stationary wave

-Thus we can say a stationary / standing wave is formed when two progressive waves of the same wavelength frequency and amplitude are made to move in opposite direction in the same medium.

-A stationary / standing wave has two important points:

i) Nodes( N) ii) Antinode (A)

NODE (N)

- This is a point of minimum disturbance in a stationary wave system

ANTINODE (A)

- This is a point of maximum disturbance in a stationary wave system

##### THE EQUATION AT STATIONARY / STANDING WAVE

– Consider two progressive waves of the same wavelength, frequency and amplitude to be moving in opposite direct within the same medium such that when they meet a stationary wave is formed.

-Let the waves be y1 and y2 such that

Y1 = A sin (

Y2 = A sin (

-According to the principle of super position of wave the resultant displacement y is given by:

Y = y1 + y2

Y = A sin (

Y = A (sin (

Y= 2A sin (

2 2

Y = 2A sin

##### PROPERTIES OF A STATIONARY/STANDING WAVE

- The distance between a node and its neighboring antinode is one quarter of a wavelength

- All point along the wave have different amplitude.
- Point between successive nodes are in phase.

##### DIFFERENCE BETWEEN PROGRESSIVE AND STATIONARY WAVES

PROGRESSIVE WAVE | STATIONARY WAVE |

1)The wave advance with a constant speed | 1) The wave does not advance but remain confined in a particular region |

2) The amplitude is the same for all the particles in the path of the wave | 2) the amplitude varies according to position being zero at the nodes and maximum at the anti nodes |

) All particles within one wavelength have difference phase | 3)phase of all particles between two adjacent nodes is the same |

4) Energy is transmitted in the direction of the wave | 4)energy is associated within the wave but there is no transfer of energy across any section of the medium |

Problem 09

- Explain the difference between progressive waves and stationary wave
- Progressive wave and stationary wave each has the same frequency of 250HZ and the same velocity of 30. Determine

(The phase difference between the vibrating points on the progressive wave which are 10cm apart

The distances between nodes in stationary wave

(iii) The equation of the progressive wave if its amplitude is 0.03m

Problem 10

Two waves travelling in opposite directions producing a standing wave. The individual functions are given by:

– Where x and y are in cm

(Find the maximum displacement of motion at

( Find the positions of nodes and antinodes

Problem 11

The transverse displacement of a string clamped at its two ends given by:

Where x, y are in metre and t is in the seconds

( Does the function represent a travelling or stationary wave?

( Interpret the wave as superposition of two waves travelling in opposite directions

What are the wavelength, frequency and speed of propagation of each wave?

Problem 12

A standing wave on the string has nodes at What is the wavelength of the travelling wave?

Problem 13

The equation of standing wave is given by

– Where x and y are in cm and t in second

Find: (Frequency and wavelength

(

SOUND WAVE

##### Definition

Sound is a longitudinal wave motion which is conveyed through an elastic medium from a vibrating body to a listener.

##### AUDIBLE RANGE

-This is a range of frequency which can be detected by human ear.

– The human ear is very sensitive and can detect very faint sounds.

-The human ear can detect sounds in the frequency range of 20HZ to 20000HZ

-With increased age upper limit of frequency range fall quite considerably

-Howev

er, the ear is most sensitive sound with frequency around 3000HZ

er, the ear is most sensitive sound with frequency around 3000HZ

INFRASONIC SOUND

-This is a sound with frequency below 20HZ.

##### ULTRASONIC SOUND

-This is sound with frequency above 20000HZ.

-Some animals including dogs, cats, bats and dolphins can detect ultrasonic sound with frequencies‘ as high as 100000HZ.

##### PROPAGATION OF SOUND WAVE

-Sound waves require a material medium to be transmitted through.

- It cannot be transmitted through a vacuum.

-Sound is transmitted by vibration of particles.

-One particle vibrates to transfer energy to the next until the sound reaches another point.

- If the particles are closer together, sound will travel faster.
- Hence sound travel with different velocities in different materials.

##### IN SOLID

-The velocity (v) of sound in the form of a rod wire depends on:

Young’s modulus E of the solid.

Density of the solid.

-One uses dimension analysis to get relationship

-Which gives

##### IN FLUIDS

-The velocity (v) of the sound in fluid (liquid gas) depends on:

() Bulk modulus

() Density of the fluid

– Again we use dimension analysis to get relationship

##### NEWTON’S FORMULA

- Newton assumed that the propagation of sound in a gas takes place under isothermal condition (constant temperature condition) in isothermal conditional Boyle`s law applies and hence: PV = K differentiating this equation:

- Where = change in pressure

- By definition:

Under isothermal condition

From the expression of velocity of sound in fluids:

Applying this equation to the case of air at S.T.P

Given

1.013 x 105Pa

V = 280ms-1

-This value is less than the actual value, which is a about 33oms-1

-This large discrepancy shows that Newton`s formula is not correct.

LAPLACE`S CORRECTION

- Laplace pointed out that Newton`s assumption of isothermal propagation is not correct.

##### REASON

-The compressions and rarefactions follow each other so rapidly that there is no time for the compressed layer (Which is at a higher temperature) and the rarefied layer (Which is at a lower temperature) to equalize their temperatures with the surroundings.

-As a result the sound propagates under adiabatic and not under isothermal conditions. LAPLACE` S EQUATION

-Under adiabatic condition of air:

Of air constant

Where

Differentiate equation (i) above

P

Under adiabatic condition

From expression of velocity of sound in

………………………………(iii)

– Applying this equation to the case of air at S.T.P.

GIVEN

V = 1.4

P = 1.013 X 105Pa

P = 1.29kgm-3

V =

1.29

V = 330ms-1 which agree with the experimental value.