Waves

What is a wave?

What is a wave?

A wave is a disturbance propagated from one

place to another with no actual transport of

matter

Classification of a wave

EM waves vs Mechanical waves

Mechanical wave

Motion requires a medium

Electromagnetic wave

Motion does not need a medium

Electromagnetic wave

Transverse wave

Displaces particles

perpendicular to the motion of the wave

Longitudinal wave

Cause the particles of a

medium to move parallel to the direction of

motion of the wave

Longitudinal wave

Has a crowded area causing a high-pressure region called

compression, and the opposite is

rarefaction

Periodic motion

The repetitive motion of a particle about the equilibrium position caused by a restoring

force when it is displaced

Simple Harmonic Motion (SHM)

• within the elastic limit, the distortion in an object is directly proportional to the distorting force

• If the restoring force obeys Hooke’s law, then the resulting vibration is called Simple Harmonic Motion (SHM)

Hooke’s law

Periodic Wave Characteristics

Wavelength (λ)The distance from crest to crest (or trough to trough); expressed

in meters

Periodic Wave Characteristics

Amplitude (A)

The distance of crest (or trough) from the midpoint of

the wave

Periodic Wave Characteristics

Frequency (f)The number of waves that

passed a fixed point per second; measured in hertz (Hz)

Periodic Wave Characteristics

Period (T)The time it takes a wave to travel a

distance equal to a wavelength; measured in second

T= 1/f

Periodic Wave Characteristics

Wave velocity (v)

Distance travelled by a wave crest in one period

v= λ/T

Periodic Wave Characteristics

Speed of a transverse wave (v)

F= tension μ= mass per unit length

v=√ (F/μ)

Periodic Wave Characteristics

Speed of a longitudinal wave (v)

B= bulk modulus of the mediumρ=density of the medium

v=√( B/ρ)

Periodic Wave Characteristics

Speed of a longitudinal wave (v)

Ƴ= Young’s modulus of the mediumρ=density of the medium

v=√ (Ƴ/ρ)

Example

The linear mass density of the clothesline is 0.250kg/m. How much tension does Rocky have to apply to produce the observed wave speed of 12.0m/s? F= 36.0N

Evaluate

Hold an alarm clock at arm’s length from your ear. While it’s still ringing , place the clock on the tabletop, press

your ear against the table. Which medium is more efficient in conducting

sound?

ExampleA hiker shouts on top of a mountain toward a vertical cliff, 688m away. The echo is heard 4s after. a) What is the speed of sound? b) The wavelength of the sound is 0.75m. What is its frequency? c) What is the period of the wave?

v=344.5m/s

f=458.67/s

T= 2.18x10-3s

remember

The displacement (y) of a wave is always a function of x and t

remember

k= wave numberk= 2π/λ

Note: Wave is moving to the +x

Power of a Wave

The rate of energy transfer (Pav)

ω= angular frequency; A= amplitude

ω=2πf

𝑃𝑎𝑣=12

√𝜇 𝐹𝜔2 𝐴2

Intensity of a Wave

The average power per unit area

For fluids in a pipe

For a solid rod

Example: Work as a group

a)In the previous example, if the amplitude of the wave is 0.075m and f=2Hz at what maximum rate does Rocky put energy into the rope? That is, what is his max. instantaneous power? b) What is his ave. power? c) As Rocky tires, the amplitude decreases. What is the average power when the amplitude has dropped to 7.50mm?

Pmax=2.83 W

Pav=1.415 W

Pav= 0.01415W

Periodic wave phenomena

Displacement y as a function of x and t

Wave reflects from a fixed end

Wave reflects from a free end

Boundary Conditions

the conditions at the end of the string, such as a rigid support or the complete

absence of transverse force

The total displacement at pt. O is zero at all times

The total displacement at pt. O is not zero but the slope is always zero.

The Principle of Superposition

When two waves overlap, the actual displacement of any point on the string at any time is obtained by adding the displacement the point would have if

only the first wave were present and the displacement it would have if only the

second wave were present.

The wave pattern doesn’t appear to be moving in either direction along the string

Nodes are points at which the string

never moves

Antinodes are points at which the amplitude of

string motion is greatest

Constructive interference

Destructive interference

Zero displacement

Large resultant displacement

The wave function for a standing wave

Sum of the individual wave function

or

Adjacent nodes are one half wavelength

apart

So the length of the string must be:

n= 1, 2, 3, …..For strings fixed at

both ends

Normal Modes on a String

N NAl

l =λ/2

λ =2l

Normal Modes on a String

N N

lAA

Nl =λ

Normal Modes on a String

N N

lA

N

l =3λ/2

NA A

λ =2l / 3

Normal Modes on a String

A A AAN N N N N

l

l =2λ

λ = l /2

Normal Modes on a String

For an oscillating system, it is a motion in which all particles of

the system move sinusoidally with the

same frequency

Fundamental frequency

The smallest frequency that corresponds to the largest

wavelength; n=1

SW frequencies

The possible standing wave frequencies

n= 1, 2, 3, …..For strings fixed at

both ends

Harmonic series

The series of possible standing wave frequencies (>f1)

Note: Musicians call these frequencies overtonesf2 is the second harmonic or the first overtone

f3 is the third harmonic or the second overtone

Standing waves and String Instruments

Shows the inverse dependency of frequency on length

Example

In an effort to get your name in the Guinness Book of World Records, you set out to build a bass viol with strings that have a length of 5.00m between fixed points. One string has a linear mass density of 49.0g/m and a fundamental frequency of 20.0Hz (the lowest frequency that the human ear can hear). Calculate (a) the tension of this string, (b) the frequency and the wavelength on the string of the second harmonic, and (c) the frequency and wavelength on the string of the second overtone.

Wave front

• It is the locus of all adjacent points on a wave that are in phase

• Phases are points on successive wave cycles of a periodic wave that are displaced from their rest position by the same amount in the same direction and are moving in the same direction

Interference

Superposition of two waves as they come in contact with

each other

Huygens Principle

States that waves spreading out from a point source may be regarded as the overlapping of tiny secondary wavelets and that every point on any wave front may be regarded as a new point source of secondary wavelets

Constructive interference Destructive interference

Two interacting waves are in-phase

Two interacting waves are out-of-phase

Diffraction

Bending of waves upon

interacting with an obstacle