Option A - Wave Phenomena

Standing Waves, Resonance, Doppler Effect, Diffraction, Resolution,

Polarization

Standing Waves

• Standing Waves are produced when interference occurs - the waves that meet:

• Have the same amplitude• Have the same frequency• Are traveling in opposite directions

Standing Waves

• Nodes - points at rest

• Antinodes - points where maximum movement takes place

• Wave pattern remains fixed in space

Sketches:

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Stationary Wave Normal (Traveling) Wave

Amplitude

Frequency

Wavelength

Phase

Energy

All points reach the same amplitude

Amplitude reached varies from 0 to A

All points oscillate with same frequency

Energy stays within wave

Energy is transmitted

All points between nodes are moving in phase

Points have different phases

Twice the distance between nodes

Shortest distance between two points in phase

All points oscillate with same frequency

Resonance• A system can be forced to vibrate at

any frequency

• System has its own natural frequency of vibration.

• Resonance occurs when system is forced to vibrate at its natural frequency - amplitude of oscillation grows and energy reaches a maximum

Boundary Conditions

• Boundary conditions at the edges of standing wave define the resonant mode of the system

Transverse waves on a string

• String is fixed at both ends, ends are nodes and cannot oscillate

• Both ends reflect, creating a standing wave

• Resonance mode with lowest frequency is called the fundamental frequency or first harmonic

• Higher resonant modes are harmonics• Example - stringed musical

instruments

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Longitudinal Sound waves in a Pipe

• Column of air enclosed in a pipe

• Boundary conditions define system

• Closed end is node, open end is antinode

• Pressure is maximum at antinodes, minimum at nodes

• Musical instruments - flute, trumpet, organ pipes

Air column closed at one end

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Air column open at both ends

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• Closed at one end - can only produce odd harmonics

• Open at both ends -can produce all harmonics

Example:

• An organ pipe (open at one end) is 1.2 meters long. Calculate the fundamental frequency. The speed of sound is 330 ms-1.

Solution:

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L =1.2m

€

λ =4L = 4.8 meters

€

f =v

λ

€

=330 ms−1

4.8 m

€

=69 Hz

The Doppler Effect

• Apparent change in frequency

• Results from relative movement between source and observer

Doppler Effect

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Red Shift/Blue Shift

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Equations:

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Δf =v

cf Where v is relative

speed between source and observer, and c is speed of light

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′ f = fv

v+us

⎛

⎝ ⎜

⎞

⎠ ⎟ moving

source

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′ f = fv+uo

v

⎛

⎝ ⎜

⎞

⎠ ⎟

moving observer

Example:

• The frequency of a car’s horn is measured by a stationary observer as 200 Hz when the car is at rest. What frequency will be heard if the car is approaching the observer at 30 ms-1? (Speed of sound in air is 330 ms-1 )

Solution:

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f = 200Hz

€

′ f = ?

€

v = 330 ms−1

€

′ f = fv

v+us

⎛

⎝ ⎜

⎞

⎠ ⎟

€

us = 30 ms−1

€

=200Hz330 ms−1

330 ms−1 − 30 ms−1

⎛

⎝ ⎜

⎞

⎠ ⎟

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=220Hz

Diffraction

• Waves pass through narrow apertures (openings) and spread out

• Extent of spreading depends on size of aperture compared to wavelength

• Narrow aperture behaves like a point source

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€

b

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b

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b

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b

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b

€

b

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θ =λb

For a single slit - the angle at the first minimum depends on wavelength and width of slit, b

Huygens’ Principle and Diffraction

bθ path difference across slit

b sinθ Each point on a wave acts as a source of a new wave

Huygens’ Principle and Diffraction

€

λ =bsinθ

€

sinθ =λ

bSince the angle is small:

€

sinθ = θ

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θ =λb

Circular Aperture

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θ =1.22λ

b

b is diameter of aperture (opening)

Angle of first minimum

Rayleigh Criterion• Two sources close together are seen as

one single source of light

• If the eye can tell the sources apart, they are resolved

• Appearance of light sources depends on diffraction

• If two sources are just resolved, the first minimum of one diffraction pattern is on top of the first maximum of the other.

Rayleigh Criteria Diagrams

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Just Resolved:

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If two sources are just resolved, the first minimum of one diffraction pattern is on top of the first maximum of the other.

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Example

Late one night, a student was approaching from a long distance away. She noticed that when she first observed the headlights of the car, they appeared to be one point of light. Later, when the car was closer, she was able to see two separate points of light. If the wavelength of the light is 500 nm and the diameter of her pupil is 4mm, calculate how far away the car was when she could first distinguish two points of light. The distance between the headlights is 1.8 m.

Solution:

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θ =1.22λ

b

€

θ =1.22(5x10−7 m)

0.004m

€

=1.525x10−4

Since angle is small:

€

θ =1.8km

x

€

=1.8km

1.5x10−4

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=11.8km

Polarization

• Polarized light - oscillating electric and magnetic fields at right angles to each other

• Transverse waves are vibrating in a “plane of vibration”

• Waves are oriented in an infinite number of ways

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• Unpolarized light - planes vary randomly

• Polarized light - fixed plane of vibration

• Light can be partially polarized, or can be circularly polarized (rotates uniformly)

• Most sources of light are unpolarized

• Radio waves, radar, microwaves are polarized

“Head on” view of polarization

PolarizedUnpolarized

Brewster’s Law

• Light is reflected and refracted

• Reflected ray can be partially polarized

• If reflected ray is completely polarized, it is perpendicular to the refracted ray

• The angle of incidence is called the polarizing angle when reflected and refracted ray are perpendicular

Brewster’s Law Diagram

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Brewster’s Law Equation

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θi + θ r = 90°

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n =sinθ i

sinθ r

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=sinθ i

cosθ i

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=tanθ

When incident medium is air:

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When incident medium isn’t air:

Malus’ Law

• Measures the reduction in intensity of light due to polarization

• Light is polarized by an analyzer

• Intensity of light is proportional to amplitude

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Iα E 2

Malus’ Law Equation

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I = I0 cos2 θ

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I = transmitted intensity in W m-2

I0 = incident intensity in W m-2

θ = angle between plane of vibration and

analyzer's preferred direction

Uses of polarization• Polarized sunglasses - reduced light

intensity due to partial horizontal polarization (axis oriented 90° to plane of reflected light), reduce glare by absorbing some reflected light

• Determining concentration of solutions -plane of polarization rotates depending on concentration of solution

• Stress analyses - polarized white light passes through plastics and glass, regions of maximum stress identified by brightly colored lines

• Liquid-crystal displays (LCD’s) - calculator displays and computer monitors - can be made to appear brighter or darker depending on degree of polarization