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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
2 Waves 11
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
3 Waves 11
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)
4 Waves 11
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
5 Waves 11
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
6 Waves 11
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
7 Waves 11
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
8 Waves 11
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)
9 Waves 11
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
10 Waves 11
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.
11 Waves 11
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
12 Waves 11
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
13 Waves 11
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
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