Development Economics

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4.1 – Oscillations

  • Simple harmonic oscillations

Oscillations are periodic motions which center around an equilibrium position.

Simple harmonic motion (SHM) is a special type of oscillation. For example:

  • The simple pendulum
  • The vibration of strings in a violin
  • The spring-mass system, where the mass is initially displaced to produce a periodic motion around the equilibrium position

An object undergoes SHM if it experiences a force which is proportional and opposite of the displacement from its equilibrium position.

 

The period is independent of the amplitude of the SHM and can be given by the following equation2-12-shm-011

for a pendulum

and

2-12-shm-0141

for a spring-mass system

 

  • Time period, frequency, amplitude, displacement and phase difference

Displacement (x) Amplitude Period (T) Frequency (f) Phase difference
Displacement of the oscillating object at a specific time from its equilibrium position Maximum displacement of the oscillating object Time taken for one complete oscillation (in seconds) Number of times the object oscillates per unit time (usually one second)

f=1/T

The difference between two SHMs with the same frequency in terms of their relative position in a cycle measured in radian

simple_harmonic_motion1

  • Conditions for simple harmonic motion

  • When the body is displaced from equilibrium, there must exist a restoring force (a force that wants to pull the body back to equilibrium).
  • The magnitude of the restoring force must be proportional to the displacement of the body and acts towards the equilibrium.

4.2 – Travelling waves

  • Travelling waves

A travelling wave is a continuous disturbance in a medium characterized by repeating oscillations. For example:

  • A rope that is flicked up and down continuously creates a repeating disturbance similar to the shape of a sine/cosine wave.

Energy is transferred by waves.

Matter is not transferred by waves.

The direction of a wave is defined by the direction of the energy transfer.

 

  • Wavelength, frequency, period and wave speed

waveparameters

Wavelength, frequency, and period follow the same rules of SHM.

Wave speed can be calculated by the following equation

velocityformula

 

  • Transverse and longitudinal waves

Transverse wave Longitudinal wave
The direction of oscillation is perpendicular to the direction of the wave2-13-waves-transverse

 

The direction of oscillation is parallel to the direction of the wave2-13-waves-longitudinal
Example:

Ÿ   Water waves

Ÿ   Wave in a string flicked up and down

Ÿ   Light (electromagnetic waves)

Example:

Ÿ   Wave produced in a spring

Ÿ   Sound waves

Ÿ   Earthquake P-waves

FYI

Transverse wave Longitudinal wave
A point with maximum positive displacement is called a crest.

A point with minimum displacement is called a trough.

A region where particles are closed to each other is called a compression.

A region where particles are furthest apart from each other is called a rarefaction.

83cd585e3e1683b350266e315ca9dc35 52e3c96ab91761a874ba06c79cce77f7

 

  • The nature of electromagnetic waves

All EM waves travel in vacuum at the same speed of 3*10^8m/s.

EM waves are transverse waves.

em_spectrum

 

  • The nature of sound waves

The speed of sound in 20 degrees Celsius dry air is approximately 343.2m/s.

Sound waves are longitudinal waves.

lwav2

4.3 – Wave characteristics

  • Wavefronts and rays

Wavefronts:

  • Lines joining points which vibrate in phase.
  • Can be straight lines or curves.
  • The distance between successive wavefronts is the wavelength of the wave.

Rays:

  • Lines which indicate the direction of wave propagation.
  • Rays are perpendicular to wavefronts.

rays

 

  • Amplitude and intensity

The amplitude and intensity of a wave depends on its energy.

The intensity of a wave is proportional to the square of its amplitude (I∝A^2).

11

  • Transverse and longitudinal waves

See previous section with the same title.

 

  • Superposition

The principle of superposition states that the net displacement of the underlying medium for a wave is equal to the sum of the individual wave displacements.

superposition

The left shows constructive interference (superposition) where the two waves add up (e.g. 1+1=2). The right shows deconstructive interference (superposition) where the two waves cancel each other (e.g. 1+(-1)=0).

 

  • Polarization

Light is a transverse wave (polarization only occur to transverse waves).

The polarization of light refers to the orientation of the oscillation in the underlying electric field.

Light is plane polarized if the electric field oscillates in one plane.

difference-between-polarized-and-unpolarized-light-how_a_polarizing_filter_works

Left shows unpolarized light and right shows polarized light.

Polarization by reflection

When light is transmitted across a boundary between two mediums with different refractive indexes, part of the light is reflected and the remaining part is refracted (for further explanation, see section 4.4).

The light reflected is partially polarized, meaning that it is a mixture of polarized light and unpolarized light.

The extent to which the reflected light is polarized depends on the angle of incidence and the refractive index of the two mediums.

The angle of incidence at which the reflected light is totally polarized is called the Brewster’s angle (ϕ) given by the equation

image001

where n1 and n2 are the refractive indexes for their respective mediums

When the angle of incidence is equal to Brewster’s angle, the reflected ray is totally polarized and the reflected ray is perpendicular to the refracted ray.

1740_brewsters20law

 

Polarizers and Analyzers

Polarizer:

  • A polarizer is a sheet of material which polarizes light.
  • When unpolarized light passes through a polarizer, its intensity is reduced by 50%.

Analyzer:

  • When polarized light passes through a polarizer, its intensity will be reduced by a factor dependent on the orientation of the polarizer. This property allows us to deduce the polarization of light by using a polarizer.
  • A polarizer used for this purpose is called an analyzer.

Malus’ Law relates the incident intensity and transmitted intensity of light passing through a polarizer and an analyzer.

1-2-polarization-a-018

where I is the transmitted intensity, I0 is the initial light intensity upon the analyzer, θ is the angle between the transmission axis and the analyzer.

1-2-polarization-019

1-2-polarization-002

When light passes through an optically active substance, the plane of polarization rotates.

 

4.4 – Wave behaviour

  • Reflection and refraction

Reflection

reflection

Angle of incidence = Angle of reflection

Reflection of waves from a fixed end is inverted.

fixedrope

Reflection of waves from a free end is not inverted.

looserope

Refraction

Refraction is the change in direction of a wave when it transmits from one medium to another.

8097906_orig

The angle of incidence and the angle of refraction can be determined by Snell’s law given by the following formula

snell2

where n1 and n2 are the refractive indexes for their respective mediums

1-2-reflectionrefraction-014

1-2-reflectionrefraction-016

Fast-to-slow: towards normal; slow-to-fast: away from normal

In addition, the refractive index n1 and n2 are related by the following equation

snells_law

where v1 and v2 are the speed of the waves in their respective mediums and λ1 and λ2 are the wavelength of the waves of their respective mediums

changingmedia-new

 

  • Snell’s law, critical angle and total internal reflection

See previous section (Reflection and refraction) for Snell’s law.

internal

The refractive index and the critical angle are related by the following equation

formula1

Total internal reflection only occurs when the light ray propagates from a optically denser medium to an optically less dense medium.

 

  • Diffraction through a single-slit and around objects

  • Diffraction through a single-slit

slitwidvar2

Single-slit equations are not required for the standard level course.

  • Diffraction around objects

difr

  • Interference patterns

Maximums form at constructive interference (the maximum is shown by 1-2) and minimums form at deconstructive interference (the first minimum is shown by 3-4).

sinslitwid

1-5-interference-007

1-5-interference-009

1-5-interference-labels-012

 

  • Double-slit interference

Like single-slit diffraction, double-slit diffraction occurs via the same methods of interference and has a similar diffraction pattern.

dslitdslit

 

  • Path difference

pathlengths

pathdifference

1-5-doubleslitrevised-009

 

4.5 – Standing waves

  • The nature of standing waves

Standing waves (stationary) waves result from the superposition of two opposite waves which are otherwise identical.

Energy is not transferred by standing waves.

u10l4b1

A wave hits a wall and is reflected identically opposite.

swf

The black wave shows the wave created by the superposition of the blue and green waves.

u10l4eani1

 

  • Boundary conditions

Air particles can oscillate and create standing waves in pipes with open or closed ends.

Antinodes are positioned at open ends and nodes are positioned at closed ends.

Standing waves on a string is equivalent to that in a pipe which is closed on both ends (nodes-node).

2-13-sound141121-017

The following table summarizes the behavior of standing waves in pipes and strings:

One closed end and one open end Two closed ends Two open ends
u11l5d1 1st Harmonic

u10l4e1

u11l5c1
u11l5c11 2nd Harmonic

u10l4e2

u11l5c2
u11l5d3 3rd Harmonic

u10l4e3

u11l5c3
nth Harmonic

λ=4L/n

(Note that even harmonics do not exist for pipes with one closed end and one open end)

nth Harmonic

λ=2L/n

nth Harmonic

λ=2L/n

 

  • Nodes and antinodes

Positions along the wave which are fixed are called nodes (minimum) and those with the largest displacement are called antinodes (maximum).

For standing waves, the distance between adjacent nodes = the distance between adjacent antinodes = λ/2.

u10l4d2ani

FYI

Difference between standing waves and travelling waves

Standing waves Travelling waves
Ÿ   No energy is propagated along a standing wave

Ÿ   A standing wave has nodes and antinodes

Ÿ   The amplitude of the standing wave varies along the wave

Ÿ   Particles between two adjacent nodes oscillate in phase and particles separated by exactly one node oscillate in antiphase.

Ÿ   Energy is propagated in a travelling wave

Ÿ   A travelling wave has neither nodes nor antinodes

Ÿ   The amplitude of the standing wave is constant throughout the wave

Ÿ   The phase difference between two particles of a travelling wave can take any value between 0 and 2π

 

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