1

can also be expressed in an equivalence form

via the relationship c = and x = ctWhere t is the time required to measure the frequency of the waveThe more we know about the value of the frequency of the wave, the longer the time taken to measure itIf u want to know exactly the precise value of the frequency, the required time is t = infinityWe will encounter more of this when we study the Heisenberg uncertainty relation in quantum physics

2 x

1 t

The classical wave uncertain relationship

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Wave can be made more ``localised’’

We have already shown that the 1-D plain wave is infinite in extent and can’t be properly localised (because for this wave, x infinity)However, we can construct a relatively localised wave by :adding up two plain waves of slightly different wavelengths (or equivalently, frequencies)

Consider the `beat phenomena’

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Two pure waves with slight difference in frequency and wave number , kkk are superimposed

,

);cos( 111 txkAy )cos( 222 txkAy

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`envelop’ (group waves)

txkk

txkkA

yyy

22cos}{}{

2

1cos2 1212

1212

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Phase waves

The resultant wave is a ‘wave group’ comprise of an `envelop’ (or the group wave) and a phase waves

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As a comparison to a plain waves, a group wave is more ‘localised’ (due to the existence of the wave envelop. In comparison, a plain wave has no `envelop’ but only ‘phase wave’)

It comprises of the group wave (the `envelop’)

moving at group velocity vg = /k

txktxkk

pp

cos22

cos 1212

tkxAtxkkA 2

1cos2}{}{

2

1cos2 1212

and the phase waves (individual waves oscillating inside the envelop)

moving at phase velocity vp = p/kp

In general, vg = /k << vp = (/(k1 + k2) because ≈, k1≈ k2

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`envelop’ (group waves). Sometimes it’s called ‘modulation’

txktkxAyyy pp

cos

2

1cos221

Phase waves

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The energy carried by the group wave is concentrated in regions in which the amplitude of the envelope is largeThe speed with which the waves' energy is transported through the medium is the speed with which the envelope advances, not the phase waveIn this sense, the envelop wave is of more ‘physical’ relevance in comparison to the individual phase waves (as far as energy transportation is concerned

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Wave pulse – an even more `localised’ wave

In the previous example, we add up only two slightly different wave to form a train of wave groupAn even more `localised’ group wave – what we call a “wavepulse” can be constructed by adding more sine waves of different numbers ki and possibly different amplitudes so that they interfere constructively over a small region x and outside this region they interfere destructively so that the resultant field approach zero

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A wavepulse – the wave is well localised within x. This is done by adding a lot of waves with with their wave parameters {Ai, ki, i} slightly differ from each other (i = 1, 2, 3….as many as it can)

such a wavepulse will move with a velocity

0kg dk

dv

(Compared to the group

velocity considered earlier vg = /k)

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Comparing the three kinds of wave

x ∞

x x

xWhich wave is the most localised?

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Why are waves and particles so important in physics?

Waves and particles are important in physics because they represent the only modes of energy transport (interaction) between two points. E.g we signal another person with a thrown rock (a particle), a shout (sound waves), a gesture (light waves), a telephone call (electric waves in conductors), or a radio message (electromagnetic waves in space).

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Interactions take place only between (i)  particles and particles (e.g. in

particle-particle collision, a girl bangs into a guy) or

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(ii)waves and particle, in which a particle gives up all or part of its energy to generate a wave, or when all or part of the energy carried by a wave is absorbed/dissipated by a nearby particle(e.g. a wood chip dropped into water, or an electric charge under acceleration, generates EM wave)

An oscillating electron generates EM waves

This is an example where particle is interacting with wave; energy transform from the electron’s K.E. to the energy propagating in the form of EM wave wave

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Waves superimpose, not collideIn contrast, two waves do not interact in the manner as particle-particle or particle-wave doWave and wave simply “superimpose”: they pass through each other essentially unchanged, and their respective effects at every point in space simply add together according to the principle of superposition to form a resultant at that point -- a sharp contrast with that of two small, impenetrable particles

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A plain 1-D EM wave (ie. a wave that propagates only in one direction)

According to Maxwell theory, light is a form of energy that propagates in the form of electromagnetic wave

Light is synonym to electromagnetic radiation

Other forms of EM radiation include heat in the form of infra red radiation, visible light, gamma rays, radio waves, microwaves, x-rays

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In the classical Maxwell theory, EM radiation is described as an oscillating electrical and magnetic fields that propagate in the form of a transverse waveit has two oscillating electric and magnetic

fields characterised by E = E0 sin(t- kx + ), B = B0 sin(t- kx + ){E,B}are orthogonal to each otherThe EM fields varies sinusoidally in time for

any fixed point x in space extends over all possible values of x The EM wave is travelling in the direction

orthogonal to the directions of the {E,B} fields (z in figure 3.1), so that {E,B,z} form a right-handed screw systemThe speed of light is related to the freq and

wavelength as per c = = k where = 2(angular frequency) k =

2(wave number), phase constant

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EM radiation transports energy

The way how wave carries energy is described in terms of ‘energy flux’, in unit of energy per unit area per unit timeAnalogous to the energy transported by a stream of water in a hoseEnergy flux of the EM wave is described by the Poynting vectorS = E x B = (E0B0/0)sin

2(kz – t + )S is perpendicular to the directions of both E and B; it describes the direction in which the energy is transportedThe rate of energy transported by the EM wave across an orthogonal area A is

z

AEc

Pave20

02

1

(in unit of energy per unit time)

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The energy flux of the light, S = Pave /A, in unit of energy per unit time per unit area

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Essentially,

Light, according to Maxwell’s EM theory, is EM wave

It display wave phenomenon such as diffraction and interference that is not possible for particles free particles only travel in straight line and they don’t bend when passing by a cornerHowever, for light, it does

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Diffraction displayed by light – a wave phenomena