Waves

Objectives• 4.4.1 Describe a wave pulse and a continuous progressive (travelling) wave.• 4.4.2 State that progressive (travelling) waves transfer energy.• 4.4.3 Describe and give examples of transverse and of longitudinal waves.• 4.4.4 Describe waves in two dimensions, including the concepts of wavefronts

and of rays.• 4.4.5 Describe the terms crest, trough, compression and rarefaction.• 4.4.6 Define the terms displacement, amplitude, frequency, period, wavelength,

wave speed and intensity.• 4.4.7 Draw and explain displacement–time graphs and displacement–position

graphs for transverse and for longitudinal waves.• 4.4.8 Derive and apply the relationship between wave speed, wavelength and

frequency.• 4.4.9 State that all electromagnetic waves travel with the same speed in free

space, and recall the orders of magnitude of the wavelengths of the principal radiations in the electromagnetic spectrum.

Wave Definition

• A wave is a disturbance (vibration) that carries energy from one point to another without the transmission of matter (particles).

• Waves which move energy from one point to another are called progressive waves.

Mechanical & Electromagnetic Waves

• Mechanical waves require a medium (air, water, string) to travel (propagate) through.

• Examples are seismic waves, sound waves and water waves.

• Mechanical waves can travel at any speed.• Electromagnetic waves do not need a medium to

travel through and can travel through a vacuum.• All EM waves travel at 3.00 * 108 m/s in a vacuum.

Mechanical Waves

• Longitudinal wave the particles of the medium vibrate parallel to the direction of wave motion.

• The direction of the vibrations of the particles in the wave is along the direction in which the energy of the wave is travelling.

• Transverse wave the particles of the medium vibrate perpendicular to the direction of wave motion.

• The particles of the medium vibrate at right angles to the direction in which the energy of the wave is travelling.

Longitudinal Waves

• Longitudinal Wave Applet

Transverse wave

Longitudinal & Transverse Waves

Parts of a Transverse Wave

Key Characteristics

• Displacement the distance of a particle on a wave from its rest position (m)

Key Characteristics

• Amplitude the maximum displacement of a particle on a wave (m)

Key Characteristics

• Wavelength the distance moved during one oscillation (cycle) of a wave, the distance between two adjacent crests or troughs [λ] (m)

Key Characteristics

• Period the time for a particle in the wave to complete one cycle or vibration [T] (s)

• Frequency the number of cycles or vibrations per unit time [f] (Hz = s-1)

Key Characteristics

• Phase difference describes the relative positions of the crests or troughs of two waves of the same frequency

• In phase when the crests and troughs of two waves are aligned

• Out of phase when a crest of one wave is aligned with the trough of another wave

In Phase & Out of Phase

Key Characteristics

• Speed = frequency X wavelength

Example

• A sound wave has a frequency of 262 Hz and a wavelength measure at 1.29 m. What is the speed of the wave? How long will it take the wave to travel the length of a football field, 91.4 m? What is the period of the wave?

Example

• A tourist shouts across a canyon and hears her echo 1.29 s later. Assuming a speed of sound of 342 m/s in air, determine the distance across the canyon.

Intensity

• The amount of energy through a unit area per unit time is the intensity of the wave.

• Intensity is proportional to – the square of the amplitude of a wave– the square of the frequency of a wave

• I α f2A2

Example

• A beam of red light has twice the intensity of another beam of the same colour. Calculate the ratio of the amplitudes of the waves.

Wave Properties

Objectives• 4.5.1 Describe the reflection and transmission of waves at a

boundary between two media.• 4.5.2 State and apply Snell’s law. • 4.5.3 Explain and discuss qualitatively the diffraction of waves at

apertures and obstacles.• 4.5.4 Describe examples of diffraction.• 4.5.5 State the principle of superposition and explain what is meant

by constructive interference and by destructive interference.• 4.5.6 State and apply the conditions for constructive and for

destructive interference in terms of path difference and phase difference.

• 4.5.7 Apply the principle of superposition to determine the resultant of two waves.

• Ripple Tank Simulation

Huygen’s Principle

• Every point on a wave front is the source of a new wave.

• These new waves are sometimes called “wavelets”.

Properties of wave motions

• Reflection– Angle of reflection =

angle of incidence– No change in λ

• Refraction– Change in direction of a

wave due to a change in speed

Properties of wave motions

• Diffraction– Spreading of a wave

around a barrier (edge) or through an opening

• Interference– The result of two

overlapping waves

REFLECTION OF 2 DIMENSIONAL WAVES

All waves reflect on meeting a barrier (echoes, reflections)

The angle of incidence ( i ) equals the angle of reflection ( r ) for waves reflecting from a barrier.

The Law for Reflection

• The angle of incidence is equal to the angle of reflection

• Also - The incident ray, the reflected ray and the normal lie on the same plane

• Use this rule for any ray or wave diagram involving reflection from any surface

• For circular waves hitting a flat reflector, the reflected waves appear to come from a source, which is the same distance behind the reflector as the real source is in front of it

• Also a line joining these 2 sources is perpendicular to the reflecting surface

O I

• If a plane wave is incident on a circular reflector then the waves are reflected so that they–Converge on a focus if the surface is

concave–Appear to come from a focus if the

surface is convex

Echos

• In the case of sound, a source of sound can be directed at a plane, solid surface and the reflected sound can be picked up by a microphone connected to an oscilloscope.

• The microphone is moved until a position of maximum reading on the oscilloscope is achieved.

• When the position is recorded it is found that again the angle of incidence equals the angle of reflection.

Reflection from a fixed end - 180 phase change

Reflection from a free end - no phase change

REFRACTION• When waves travel from one medium to

another they bend.• This bending or change of direction that

occurs as a wave travels from one medium into another, called refraction, occurs because of a change in speed.

• When waves travel from one medium to another, their wave speed and wavelength change, but frequency remains constant (f1=f2)

Air

Water

Refraction occurs because light travels at different speeds in different media. The more optically dense, the slower light is.

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Shallow water

Deep water

Refraction occurs because light travels at different speeds in different media.

Medium 1

v1

Medium 2

v2

The greater the change in velocity the greater the bending of the waves.

sin

sin

i

R

v

v

f

f 1

2

1 1

2 2

But, frequency does not change, therefore f1=f2

sin

sin

i

R

v

v 1

2

1

2

Waves travel from deep water to shallow water in a ripple tank. The deep water waves have a wavelength of 5 cm and a wave speed of 10 cm/s. If the speed of the waves in the shallow water is 8 cm/s, find the new wavelength.

v

v1

2

1

2

v cm s1 10 /

1 5 cm

v cm s2 8 / 10

8

5

2

2

5 8

104

xcm

Water waves in a ripple tank enter a shallow region at an angle of incidence of 300 from a deep region. Calculate the angle of refraction if the wave speed in the deep region is 10 cm/s and the wave speed in the shallow region is 6 cm/s

v cm s1 10 /

v cm s2 6 /

sin

sin

i

R

v

v 1

2i o30

sin

sin

30 10

6R

sin sinR 6

1030

R o17 5.

SNELL’S LAW

• As light travels from one medium to another the ratio of sin i to sin R is a constant for any two medium

sin

sin

i

Rslope of graph constant =

The constant is known as the refractive index, and is a measure of the bending power of materials.

sini

SinR0·1 0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1

0·1

0·2

0·3

0·4

0·5

0·6

0·7

0·8

0·9

1

Sini Vs sinR

x

x

x

x

x

Line of best fit

Can be worked out mathematically(or Excel), but manually draw a line that best fits the points. The same distance vertically for points above and below the line.

RISE=0.65

RUN=0.4

.).2(6.1625.140.0

65.0fs

Run

RiseSlope

• It is found the refractive index is equal to the ratio of the velocities in the two medium.

Medium 1

v1

Medium 2

v2

sin

sin

i

R

v

v 1

2

The greater the change in velocity the greater the bending of light.

• As the velocity difference between the two materials increases the amount of bending increases.

45

Air

Vair=3.0 x 108 m/s

sin isin R

sin

sin

i

R

v

v 1

2

=

v1

v2

R =

Vwater= 2.25 x 108 m/s

Water32

32

Vperspex=2.0 x 108 m/s

28

28

Perspex

Vperspex=2.0 x 108 m/s

Vdiamond=1.24 x 108 m/s

Vwater= 2.25 x 108 m/s

17

Vdiamond=1.24 x 108 m/s

17Diamond

Absolute refractive index• To make comparisons of the bending power of

mediums some absolute refractive index is needed.

• This is done by using a vacuum as our reference and saying it has a refractive index of 1.0000

• We consider light coming from a vacuum into the medium.

• The absolute refractive index of a medium is equal to the ratio of the velocity of light in a vacuum divided by the velocity of light in the medium.

Vacuum

c

Medium

vm

Absolute

Refractive =index

c

vmm

C= speed of light in a vacuum

The greater the bending power of a medium then the greater the absolute refractive index of the medium.

Vwater= 2.25 x 108 m/s

Vperspex=2.00 x 108 m/s

Vdiamond=1.24 x 108 m/s

The absolute refractive index of a medium will be greater if it slows down light more

C= 3.00 x 108 m/s= speed of light in a vacuun

waterwater

c

v

x

x

300 10

2 25 10133

8

8

.

..

perspexperspex

c

v

x

x

3 00 10

2 00 10150

8

8

.

..

diamonddiamond

c

v

x

x

3 00 10

124 102 42

8

8

.

..

Medium 1

v1

Medium 2

v2

sin

sin

i

R

v

v

cv

cv 1

2

1

2

c

v

v

c2

1 c

v cv

2

1

1

2

1

sin

sin

i

R

v

v 1

2

2

1

Ri sinsin 21 We can put this in the form

Medium 1

v1

Medium 2

v2

QUESTION: Light strikes a container of water at an angle of 40o to the surface of the water. The absolute refractive index of air is 1.00 and the absolute refractive index of water is 1.33. Find

(a) the angle of refraction

(b) the angle of deviation

(c) the speed of light in water

REFRACTION EFFECTS

The bent pencil

We can see the sun after it has actually set

EARTH

SUN

increasing density

R

O

Y

G

B

I

V

DISPERSIONDispersion occurs because each different colour slows down by a different amount in the glass prism. Red is slowed down least and therefore refracted least while violet is slowed down the most and so is refracted the most.

Diffraction

Diffraction

Diffraction

Diffraction

• In general (i.e. any type of wave), the degree of diffraction depends upon:– The size of the obstacle or aperture.– The wavelength of the wave

• The greatest diffraction effects occur when the wavelength equals the size of the aperture.

• λ= a• “a” size of aperture (“distance between slits”)

Single-slit diffraction

Superposition

Superposition

• The principle of superposition states that, when 2 or more waves meet at a point, the resultant displacement at that point is equal to the sum of the displacements of each wave at that point.

• The displacement of waves can be added.

Superposition

• 2 waves are in phase & meet at a point constructive interference.

• The waves interfere constructively.

• 2 waves are out of phase & meet at a point destructive interference.

• The waves interfere destructively.

Constructive & Destructive Interference

Path Difference

Path difference

• Constructive interference path difference is λ

• Destructive interference path difference is ½ λ

Interference pattern

Interference pattern

Interference pattern

Example

• Two point sources, 3.0 cm apart, are generating periodic waves in phase. A point on the third destructive line of the wave pattern is 10 cm from one source and 8.0 cm from the other source. Determine the wavelength of the waves

Simple Harmonic Motion

Oscillations

Objectives

• 4.1.1 Describe examples of oscillations.• 4.1.2 Define the terms displacement, amplitude,

frequency, period and phase difference.• 4.1.3 Define simple harmonic motion (SHM) and state the

defining equation as a = −w2x . • 4.1.4 Solve problems using the defining equation for SHM.• 4.1.5 Apply the equations for velocity and displacement as

solutions to the defining equation for SHM.• 4.1.6 Solve problems, both graphically and by calculation,

for acceleration, velocity and displacement during SHM.

• http://www.phy.cuhk.edu.hk/phyworld/resources/java/shm/SimpleHarmonic.html

• http://www.acoustics.salford.ac.uk/feschools/waves/shm.htm

Definitions

• Oscillations one complete cycle• Period (T) time for one cycle• Frequency (f) # of cycles per unit time• Displacement (x) distance from equilibrium

position• Amplitude maximum displacement• Simple harmonic motion common type of

oscillation with displacement that is sinusoidal in nature

Simple Harmonic Motion• The motion of a particle

about a fixed point such that its acceleration ‘a’ is proportional to its displacement x from the fixed point, and is directed towards the point.

• a α x• a = -ω2x• ω2 constant that is always

positive• ω=2πf angular frequency

Solutions for SHM

• x = x0sinωt

• x = x0cosωt

• Use sin eqn for when, at t = 0, the particle is at its equilibrium position (x=0)

• Use cos eqn for when, at t =0, the particle its maximum displacement (x=x0)

Solutions for SHM

• v = x0 ωcos ω t when x = x0sinωt

• v = -x0 ωsinωt when x = x0cosωt

• Max speed at v0 = x0ω

• Alternate eqn

Solutions for SHM

• a = -x0ω2 sinωt when x = x0sinωt

• a = -x0ω2 cosωt when x = x0cosωt

X vs V vs A for SHM

Example

• An air-track cart attached to a spring completes one oscillation every 2.4 s. At t = 0, the cart is released from rest at a distance of 0.10 m from its equilibrium position. What is the position of the cart at (a) 0.30 s, (b) 0.60 s, (c) 2.7 s, and (d) 3.0 s?

Example

• Using the information from the previous example, find the velocity and acceleration of the cart at (a) 0.30 s and (b) 0.60 s.

Examples of SHM

• Two major scenarios– Mass on a helical spring– Simple pendulum

Mass on a helical spring

• According to Hooke’s law, the mass experiences a restoring force when it is displaced from equilibrium

• The acceleration causing this force gives the condition for SHM

• As a result F = -kx = ma• But a = -ω2x and ω=2π/T• Therefore, T = 2π√m/k

Example

• A 0.120 kg mass attached to a spring oscillates with an amplitude of 0.0750 m and a maximum speed of 0.524 m/s. Find (a) the spring constant and (b) the period of motion.

The simple pendulum

• For small amplitude oscillations, the pendulum bob oscillates with SHM

• The component of the restoring force (gravity) is provided by the acceleration

• Derivation gives:

• Where l is the length of the pendulum

Example

• Would it be practical to make a pendulum with a period of 10.0 s?

Example

• On a planet with an unknown value of g, the period of a 0.65 m long pendulum is 2.8 s. What is g for this planet?

Objectives

• 4.2.1 Describe the interchange between kinetic energy and potential energy during SHM.

• 4.2.2 Apply the expressions for the kinetic energy of a particle undergoing SHM, for the total energy and for the potential energy.

• 4.2.3 Solve problems, both graphically and by calculation, involving energy changes during SHM.

Energy in SHM

• Using the alternate equation for speed from before, we can derive an expression for the kinetic energy of a particle oscillating in SHM.

Energy in SHM

• Analyzing the work done by the restoring force in SHM, and using the equation for acceleration, the potential energy in SHM can be found.

Energy in SHM

• Through conservation of energy and the fact that the mass, the angular frequency and the amplitude are all constant, the total energy is constant and expressed as:

Example

• A 1.8 kg mass attached to a spring oscillates with an amplitude of 7.1 cm and a frequency of 2.6 Hz. What is its energy of motion?

Objectives

• 4.3.1 State what is meant by damping. • 4.3.2 Describe examples of damped oscillations..• 4.3.3 State what is meant by natural frequency of

vibration and forced oscillations.• 4.3.4 Describe graphically the variation with forced

frequency of the amplitude of vibration of an object close to its natural frequency of vibration.

• 4.3.5 State what is meant by resonance.• 4.3.6 Describe examples of resonance where the effect

is useful and where it should be avoided.

Free & Damped Oscillations• Free oscillations A particle is

said to be undergoing free oscillations when the only external force acting on it is the restoring force

• Free oscillations have constant amplitude and the total energy remains constant

• Damped oscillations Oscillations are damped in real life as a result of energy dissipation (conversion into heat) due to friction or other resistive forces.

Damped Oscillations

• Light damping (underdamping) The amplitude of oscillations decreases with time (exponentially)

• Critical damping If damping is increased to a point where the displacement reaches zero in the shortest possible time without any oscillations

• Overdamping If damping is increased further, the displacement decreases to zero in a longer time than critical damping

Damping

Uses of damping

• Shock absorbers (suspension in vehicles)

• Swinging doors• Many, many more

Forced oscillations & resonance

• Natural frequency Just like waves, every object going through free oscillations vibrates with a natural frequency

• If this same object is under the effect of a forced vibration, then the driving frequency can be chosen to be anything

• If the driving frequency is matched to the natural frequency, then resonance occurs

• Resonance is the frequency at which the object oscillates (vibrates) at maximum amplitude

Resonance curve (without damping)

Resonance curve (with damping)

Resonance curve (with damping)

• The amplitude of oscillation at all frequencies is reduced

• The frequency at maximum amplitude shifts gradually towards lower frequencies

• The peak becomes flatter

Barton’s pendulum

Examples of resonance

• Swings• Tacoma Bridge• Loudspeakers• Musical instruments• Car engines & car mirrors