1. 1 Waves 11 Lecture 11 Dispersive waves. D Aims: 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. 2 Waves 11 Wave groups D Packets. ëThe perfect harmonic plane wave is an idealisation with little practical significance. ëReal wave systems have localised waves - wave packets. ëInformation in wave systems can only be transmitted by groups of wave forming a packet. D Non-dispersive waves: ëAll waves in a group travel at the same speed. D Dispersive waves: ëWaves travel at different speeds in a group. D Superposition of 2 waves.  With slightly different frequencies:  ± . ëReal part is Envelope Short period wave

3. 3 Waves 11 Superposition: two frequencies Speed the envelope moves   0.05. ëBoth modulating envelope and short-period wave have the form for travelling waves. ëThey DO NOT necessarily travel at the same speed D Group velocity  Group velocity =v g =  /  k. D Phase velocity  Phase velocity = v p =  /k. Speed of the short-period wave (carrier)

4. 4 Waves 11 D Note: ëGroup velocity is the speed of the modulating envelope (region of maximum amplitude). Energy in the wave moves at the Group velocity. D General wavepacket (of any shape): ëPhase velocity: ëGroup velocity: ëEqual for a non-dispersive wave. ëOtherwise: Wave groups Energy localised near maximum of amplitude Must know

5. 5 Waves 11 Water waves D Simple treatment: ëGravity - pulls down wave crests. ëSurface tension - straightens curved surfaces. D Surface tension waves (ripples)  Important for  < 20mm. (Ignore gravity)  Dimensional analysis gives us the relation between v p and v g.  Surface tension  ; density  ; wavelength. soLT -1 =[MLT -2 L -1 ]  [ML -3 ]  [L]   Equating coefficients T: -1 = -2  so  = 1/2 M: 0 =  +  so  = -1/2 L: 1 = -3  +  so  =-1/2  An example of anomalous dispersion v g >v p. ëCrests run backwards through the group).

6. 6 Waves 11 Water waves D Gravity waves  Similar analysis for  >> 20mm and for deep water  depth (ignore surface tension).  Dimensional analysis gives us the relation between v p and v g.  Surface tension  ; density  ; wavelength. gives (the constant is unity)  An example of normal dispersion v g <v p. ëCrests run forward through the group. D Dispersion relation

7. 7 Waves 11 Guided waves ëE.g. optical fibres, microwave waveguides etc. D Guided waves on a membrane. D Guided waves on a membrane 2-D example. ëRectangular 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, k y is fixed. k x follows from  and applying Pythagoras’ theorem to k.

8. 8 Waves 11 Dispersion relation D Wave vector  k is the wavevector and v the speed for unguided waves on the membrane; i.e. ëThus ëWave velocity: Phase velocity: Dispersion relation,  =  ( k )

9. 9 Waves 11 Group velocity  Group velocity follows from differentiating  (k).  Using expression for  2 (previous overhead). ëThus,  In the present case there is a simple connection between v p and v g, which follows from [8.4].

10. 10 Waves 11 Properties of guided waves D Allowed modes ëThere is a series of permitted modes, corresponding to different n. D Wavlength  k x <k so: Wavelength of the guided wave, x, is longer than that of unguided wave,. D Wave velocity  Phase velocity exceeds speed of unguided waves. v p >v. ëGroup velocity is less than unguided wave.  v g v p =v 2.  As k x  0. v p . Note, no violation of Special Relativity since energy is transmitted at v g.  In the large k limit, behaviour approaches that of an unguided wave D Cut-off frequency  No modes with real k for  <  v/b. This is the cut- off frequency. Below this, k x 2 <0 and the wave is evanescent.

11. 11 Waves 11 Visualising the modes D n=1 (surface plot) D n=2 (surface plot) (contour plot) + + + + - - - - y x y x  y x 

12. 12 Waves 11 Evanescent waves D Below the cut-off frequency ëIn the guide, ëbelow the cut-off frequency,  k x 2 is negative, so with a real. ëThe wave has the form: ëNot oscillatory in the x-direction. ëAn evanescent wave. Oscillates with t

13. 13 Waves 11 Total internal reflection D Refraction: Snell’s Law  When sin  1 >n 2 /n 1 then sin  2 >1 !! ëThe light undergoes total internal reflection. ëAn evanescent wave is set-up in region 2.  If boundary is parallel to the y -axis:  If sin  2 >1 then Evanescent region