1 Waves 11

Lecture 11Lecture 11

Dispersive waves.Dispersive waves.

Aims:Aims:Dispersive waves.Dispersive waves.

Wave groups (wave packets) Superposition of two, different

frequencies. Group velocity. Dispersive wave systems Gravity waves in water. Guided waves (on a membrane).

Dispersion relations Phase and group velocity

+ +

++-

-

-

-

y

x

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Wave groupsWave groups

Packets.Packets. The perfect harmonic plane wave is an

idealisation with little practical significance. Real wave systems have localised waves -

wave packets. Information in wave systems can only be

transmitted by groups of wave forming a packet.

Non-dispersive waves:Non-dispersive waves: All waves in a group travel at the same speed.

Dispersive waves:Dispersive waves: Waves travel at different speeds in a group.

Superposition of 2 waves.Superposition of 2 waves. With slightly different frequencies: ±.

Real part is

)(

)..()..()(

))()(())()((

)..cos(2 kxti

xktixktikxti

xkktixkkti

exktA

eeAe

AeAe

)cos()..cos(2 kxtxktA

EnvelopeEnvelope Short period waveShort period wave

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Superposition: two frequenciesSuperposition: two frequencies

Speed the envelope movesSpeed the envelope moves

0.050.05..

Both modulating envelope and short-period wave have the form for travelling waves.

They DO NOT necessarily travel at the same speed

Group velocityGroup velocity

Group velocity =vg=/k.

Phase velocityPhase velocity

Phase velocity = vp=/k.Speed of the short-period wave (carrier)Speed of the short-period wave (carrier)

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ooop kv

o

dkdv og

Note:Note: Group velocity is the speed of the modulating

envelope (region of maximum amplitude). Energy in the wavemoves at theGroup velocity.

General wavepacket (of any shape):General wavepacket (of any shape):

Phase velocity:

Group velocity:

Equal for a non-dispersive wave. Otherwise:

Wave groupsWave groups

Energy localised nearmaximum of amplitude

Energy localised nearmaximum of amplitude

Must knowMust know

d

dvvv

kddk

kdkd

d

dv

dk

dvdk

dvkv

dk

kvd

dkd

vkv

ppg

pp

pp

pgp

22

/2;

)(;

so

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Water wavesWater waves

Simple treatment:Simple treatment: Gravity - pulls down wave crests. Surface tension - straightens curved surfaces.

Surface tension waves (ripples)Surface tension waves (ripples) Important for < 20mm. (Ignore gravity) Dimensional analysis gives us the relation

between vp and vg.

Surface tension ; density ; wavelength .

so LT-1=[MLT-2L-1][ML-3] [L]

Equating coefficientsT: -1 = -2 so = 1/2M: 0 = + so = -1/2L: 1 = -3 + so =-1/2

An example of anomalous dispersion vg>vp.

Crests run backwards through the group).

pv

pg

pp

vCkdkd

vCk

Ckk

vv

23

23 2/12/3

2/12/1

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Water wavesWater waves

Gravity wavesGravity waves Similar analysis for >> 20mm and for deep

water depth (ignore surface tension). Dimensional analysis gives us the relation

between vp and vg.

Surface tension ; density ; wavelength .

gives

(the constant is unity)

An example of normal dispersion vg<vp.

Crests run forward through the group.

DispersionDispersionrelationrelation

gvp

pg

pp

vkg

dkd

vgk

kg

kvgv

21

21

;

;

2/12/1

2/12/1

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Guided wavesGuided waves

E.g. optical fibres, microwave waveguides etc.

Guided waves on a membrane Guided waves on a membrane 2-D example. . Rectangular membrane stretched, under

tension T, clamped along edges.

Travelling wave in the x-direction.Standing wave in the y-direction.

Boundary conditions =0 at y=0 and y=b.

Thus, ky is fixed. kx follows from and applying Pythagoras’ theorem to k.

xktiykiA

ykxktiA

ykxktiA

xyBA

yxB

yxA

expsin2

exp

exp

b

nkbk yy

0sin

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Dispersion relationDispersion relation

Wave vectorWave vector

k is the wavevector and v the speed for unguided waves onthe membrane; i.e.

Thus

Wave velocity: Phase velocity:

2

2222

vkkk yx

// 222 Tkv

2

22222

2

22

2

2

2

2222

b

nkv

b

n

vb

nkk

x

x

2

22

2

2/

b

n

vkv

xp

Dispersion relation, =(k)Dispersion relation, =(k)Dispersion relation, =(k)Dispersion relation, =(k)

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Group velocityGroup velocity

Group velocity Group velocity follows from differentiating follows from differentiating ((kk)).. Using expression for 2 (previous overhead).

Thus,

In the present case there is a simple connection between vp and vg, which follows from [8.4].

4.8

22

2

2

x

xg

xx

kv

dkd

v

kvdkd

2

22

2

22

b

n

v

vvg

2

2 /

vvv

vvv

pg

pg

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Properties of guided wavesProperties of guided waves

Allowed modesAllowed modes There is a series of permitted modes,

corresponding to different n.

WavlengthWavlength kx<k so: Wavelength of the guided wave, x, is

longer than that of unguided wave, .

Wave velocityWave velocity Phase velocity exceeds speed of unguided

waves. vp>v.

Group velocity is less than unguided wave. vgvp=v2.

As kx 0. vp . Note, no violation of Special Relativity since energy is transmitted at vg.

In the large k limit, behaviour approaches that of an unguided wave

Cut-off frequencyCut-off frequency No modes with real k for <v/b. This is the cut-

off frequency. Below this, kx2<0 and the wave is

evanescent.

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Visualising the modesVisualising the modes

n=1 (surface plot) n=1 (surface plot)

n=2 (surface plot)n=2 (surface plot)

(contour plot)(contour plot)

+ +

++-

-

-

-

y

x

y

x

y

x

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Evanescent wavesEvanescent waves

Below the cut-off frequencyBelow the cut-off frequency In the guide,

below the cut-off frequency,

kx2 is negative, so with a real.

The wave has the form:

Not oscillatory in the x-direction. An evanescent wave.

)(expsin xktiykA xy bvnc /

iakx

xtiykA y expexpsin

Oscillates with tOscillates with t

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Total internal reflectionTotal internal reflection

Refraction: Snell’s LawRefraction: Snell’s Law

When sin1>n2/n1 then sin2>1 !!

The light undergoes total internal reflection. An evanescent wave is set-up in region 2. If boundary is parallel to the y-axis:

If sin2>1 then

1

2

2

1sinsin

nn

222222

111111

sincos

sincos

kkkk

kkkk

yx

yx

2 Region

1 Region

ikkk x 22

2222 sin1cos

Evanescent regionEvanescent regionEvanescent regionEvanescent region