Optics

Rayoptics

Chapter 2Wave Motion

Lecture 2

Introduction to waves Wave equation Harmonic waves

One dimensional wave

Classical traveling wave: self-sustaining disturbance of a medium, which moves through space transporting energy and momentum.

Example: sound waves

Longitudinal waves:the medium is displaced in the direction of motion

Transverse waves:the medium is displaced in the direction perpendicular to motion

Note: disturbance advances, not matter

One dimensional waves

Human wave

A typical human wave has a phase velocity of about 20 seats per second.

One dimensional wave: math

Disturbance must be a function of position and time:

txftx ,,

Shape of disturbance at any instant represents the profile of the wave:

)(0,,0

xfxftxt

at time = 0

Special case:the shape of wave does not change in time

One dimensional wave: math

Assume:- wave moves at speed v- at time t=0 its profile is f(x)

At time t the disturbance moved distance vt along x axis, but its shape is the same:

txftx v,

If we have a snapshot of a wave shape at time zero we can find a full time-dependent equation of the wave.

2exp axxf ‘Gaussian function’

2exp txa v

txftx v,What is this?:Regardless of shape, variables x and t must appear as a single unit (x±vt)

Differential wave equationFix time in equation for wave (x,t=const)

- get shape of the wave in spaceFix x in equation for wave (x=const,t)

- get dynamics of disturbance at particular coordinate

Variables x and t are independent, but appear as (x±vt) There should be a connection between the shape in space and dynamics in time.

To relate the space and time dependencies of (x,t) we will consider partial derivatives - derivatives of the function in respect to only one coordinate while the other is fixed

This approach was introduced in 18th century (Jean Le Rond d’Alembert)

v

tx'1'

xx

Differential wave equation

)'(, xftx Wave: , where txx v'

Partial derivative at fixed time:

xx

xf

xf

x

'

'

'xf

x

Partial derivative at fixed x:

tx

xf

t

'

'

'xf

t

v

xt

v The rate of change of with t and with x are

equal, to within a multiplicative constant

Differential wave equation

Second partial derivatives:

2

2

2

2

'xf

x

2

22

2

2

'xf

t

v

Differential wave equation

2

2

22

2 1tx

v

homogeneous, linear, second-order differential equation - is in every term- if is solution, then N is also

any integer

This describes ‘undamped’ system, i.e. it has no sources of damping (energy loss) as the wave propagates - the shape of wave does not change as it propagates.

For damped system: add term /t

propagationnumber

amplitude

Harmonic waves(sinusoidal waves: described by sin or cos)

Note: any wave can be synthesized as a superposition of harmonic waves

Chose profile at time zero: )()sin(,0

xfkxAxtxt

Replace x x-vt to get progressive wave traveling right at speed v:

)(sin, txkAtx v

This is sinusoidal disturbance both in time and space (the wave is periodic in space and time)

One dimensional waves

Harmonic waves: wavelength )(sin, txkAtx v

Space period (wavelength): distance between two maxima (minima).If we increase/decrease x by the disturbance should not change:

txtx ,, )(sin txkA v )(sin txkA v ktxkA )(sin v

ktxktxk )(sin)(sin vv

That is only true if 2k

2kPropagation number and wavelength are connected

)sin(, Atx Can rewrite:

phase

)( txk v

Harmonic waves: temporal shape

)2/(sin2/,2/

tkAtftxx

vv

Example: consider temporal behavior of disturbance at x=/2

t

period

period

v

Harmonic waves: period, frequencyTemporal period (): time between appearance of two maxima (minima).If we increase/decrease t by the disturbance should not change:

txtx ,,

1

)(sin txkA v )(sin txkA v vv ktxkA sin

vvv ktxktxk )(sin)(sin

Therefore: 2vk 2vk

2k

22

v

v

Frequency is number of oscillations per unit time, since one oscillation occurs in time :

combine v

angular temporal frequency: 22

wave number (spatial frequency): 1

Harmonic waves: summary

1

v

)(sin txkA v

2k

Functional shape: Wave parameters:k - propagation number - wavelength - period - frequency - angular temporal frequency - wave number

v

Alternative forms:

txA 2sin

txA 2sin

tkxA sin

txA v

2sin

“-” for wave moving right“+” for wave moving left

mostlyused

These eq-ns describe an infinite monochromatic (monoenergetic) wave.Real waves are not infinite and can be described by superposition of harmonic waves. If frequencies of these waves cluster closely to a single frequency (form narrow band) the wave is called quasimonochromatic

single frequency

22

1

Periodic waves

- wavelength - the length of one profile-element - period - the duration in time for one profile-element - wave number - number of profile-elements per unit length- etc…

Waveform produced by saxophone:

profile-elements - when repeated can reproduce the whole waveform

Can use the same parameters to describe:

Harmonic waves: example

1. Write an equation of a “red” light wave that propagates along x axis (at speed of light c) and has a wavelength 600 nm.

Solution:)(sin txkA v 2k

)(m106

2sin 7 txA c

2. What is the frequency of this light?

Solution: v

cv

Hz105m10600

m/s 103 149

8

Hz1/s

Harmonic wave: Initial phase

tkxAtx sin,Consider wavetkx phase:

When written like that it implies that 0,00

txtx

With a single wave we can always chose x axis so that above is trueBut in general case 0,

00

txtx

x

This is equivalent to the shift of coordinate x by some value a a

taxkAtx sin, katkxAtx sin,

tkxAtx sin, - initial phase

tkxphase:

Harmonic wave: Phase

x

tkxA sinCan use cos():

tkxAtx sin, 2cos, tkxAtx

equivalent equations

Special case: = = 180o phase shift

x

tkxAtx sin,

kxtAtx sin,

2/cos, kxtAtx

Note: sin(kx-t) and sin(t-kx) both describe wave moving right, but phase-shifted by 180 degrees ().

Harmonic wave: Phase derivatives

tkx Phase:

Partial derivatives:

xtrate of change of phase with time is equal to angular frequency (=2)

kx t

rate of change of phase with distance is

equal to propagation number

tkxAtx sin,

phase velocity of a wave

Harmonic wave: Phase velocity tkx Phase:

What is the speed of motion of a point with constant phase?

v

kxt

tx

t

x

from the theory of partial derivatives

sign gives direction

In general case, for any wave we can find the phase velocity:

t

x

xt

v

always >0by definition Add sign to give direction:

+ in positive x direction- in negative x direction

Phase (red) vs. group (green) velocity(to be discussed later)