Dispersive waves.

1
Lecture 11

• Dispersive waves.
• AimsDispersive 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

2
Wave groups

• 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
• All waves in a group travel at the same speed.
• Dispersive waves
• Waves travel at different speeds in a group.
• Superposition of 2 waves.
• With slightly different frequencies wDw.
• Real part is

Envelope
Short period wave
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Superposition two frequencies

• Dw/wDk/k 0.05.
• Both modulating envelope and short-period wave
  have the form for travelling waves.
• They DO NOT necessarily travel at the same speed
• Group velocity
• Group velocity vgDw/Dk.
• Phase velocity
• Phase velocity vpw/k.

Speed the envelope moves
Speed of the short-period wave (carrier)
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Wave groups

• 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)
• Phase velocity
• Group velocity
• Equal for a non-dispersive wave.
• Otherwise

Energy localised nearmaximum of amplitude
Must know
5
Water waves

• Simple treatment
• Gravity - pulls down wave crests.
• Surface tension - straightens curved surfaces.
• Surface tension waves (ripples)
• Important for l lt 20mm. (Ignore gravity)
• Dimensional analysis gives us the relation
  between vp and vg.
• Surface tension s density r wavelength
  l.so LT-1MLT-2L-1aML-3 bL g
• Equating coefficientsT -1 -2a so a
  1/2M 0 ab so b -1/2L 1 -3b g
  so g -1/2
• An example of anomalous dispersion vggtvp.
• Crests run backwards through the group).

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Water waves

• Gravity waves
• Similar analysis for l gtgt 20mm and for deep water
  lltlt depth (ignore surface tension).
• Dimensional analysis gives us the relation
  between vp and vg.
• Surface tension s density r wavelength
  l.gives (the constant is unity)
• An example of normal dispersion vgltvp.
• Crests run forward through the group.
• Dispersionrelation

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Guided waves

• E.g. optical fibres, microwave waveguides etc.
• 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 Y0 at y0 and yb.
• Thus, ky is fixed. kx follows from w and
  applying Pythagoras theorem to k.

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Dispersion relation

• Wave vector
• k is the wavevector and v the speed for unguided
  waves onthe membrane i.e.
• Thus
• Wave velocity Phase velocity

Dispersion relation, ww(k)
Dispersion relation, ww(k)
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Group velocity

• Group velocity follows from differentiating w(k).
• Using expression for w2 (previous
  overhead).
• Thus,
• In the present case there is a simple connection
  between vp and vg, which follows from 8.4.

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Properties of guided waves

• Allowed modes
• There is a series of permitted modes,
  corresponding to different n.
• Wavlength
• kxltk so Wavelength of the guided wave, lx, is
  longer than that of unguided wave, l.
• Wave velocity
• Phase velocity exceeds speed of unguided waves.
  vpgtv.
• Group velocity is less than unguided wave.
• vgvpv2.
• 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 frequency
• No modes with real k for wltpv/b. This is the
  cut-off frequency. Below this, kx2lt0 and the wave
  is evanescent.

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Visualising the modes

• n1 (surface plot)
• n2 (surface plot)(contour plot)

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Evanescent waves

• Below 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.

Oscillates with t
13
Total internal reflection

• Refraction Snells Law
• When sinq1gtn2/n1 then sinq2gt1 !!
• The light undergoes total internal reflection.
• An evanescent wave is set-up in region 2.
• If boundary is parallel to the y-axis
• If sinq2gt1 then

Evanescent region
Evanescent region