Lecture 11 Dispersive waves D Aims Dispersive waves

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 1 Waves 11

Wave groups D Packets. ë The perfect harmonic plane wave is an idealisation with little practical significance. ë Real wave systems have localised waves wave packets. ë Information in wave systems can only be transmitted by groups of wave forming a packet. D Non-dispersive waves: ë All waves in a group travel at the same speed. D Dispersive waves: ë Waves travel at different speeds in a group. D Superposition of 2 waves. ë With slightly different frequencies: w±Dw. ë Real part is Envelope Short period wave 2 Waves 11

Superposition: two frequencies D Dw/w=Dk/k= 0. 05. ë Both modulating envelope and short-period wave have the form for travelling waves. ë They DO NOT necessarily travel at the same speed Speed the envelope moves D Group velocity ë Group velocity =vg=Dw/Dk. D Phase velocity ë Phase velocity = vp=w/k. Speed of the short-period wave (carrier) 3 Waves 11

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

Water waves D Simple treatment: ë Gravity - pulls down wave crests. ë Surface tension - straightens curved surfaces. D Surface tension waves (ripples) ë Important for l < 20 mm. (Ignore gravity) ë Dimensional analysis gives us the relation between vp and vg. ë Surface tension s; density r; wavelength l. so LT-1=[MLT-2 L-1]a[ML-3] b[L] g ë Equating coefficients T: -1 = -2 a so a = 1/2 M: 0 = a+b so b = -1/2 L: 1 = -3 b + g so g =-1/2 ë ë An example of anomalous dispersion vg>vp. ë Crests run backwards through the group). 5 Waves 11

Water waves D Gravity waves ë Similar analysis for l >> 20 mm and for deep water l<< depth (ignore surface tension). ë Dimensional analysis gives us the relation between vp and vg. ë Surface tension s; density r; wavelength l. gives (the constant is unity) ë An example of normal dispersion vg<vp. ë Crests run forward through the group. D Dispersion relation 6 Waves 11

Guided waves ë E. g. optical fibres, microwaveguides etc. D Guided waves on a membrane 2 -D example. ë Rectangular membrane stretched, under tension T, clamped along edges. ë Travelling wave in the x-direction. Standing wave in the y-direction. ë Boundary conditions Y=0 at y=0 and y=b. ë Thus, ky is fixed. kx follows from w and applying Pythagoras’ theorem to k. 7 Waves 11

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

Group velocity D Group velocity follows from differentiating w(k). ë Using expression for w 2 (previous overhead). ë Thus, ë In the present case there is a simple connection between vp and vg, which follows from [8. 4]. 9 Waves 11

Properties of guided waves D Allowed modes ë There is a series of permitted modes, corresponding to different n. D Wavlength ë kx<k so: Wavelength of the guided wave, lx, is longer than that of unguided wave, l. D Wave velocity ë Phase velocity exceeds speed of unguided waves. vp>v. ë Group velocity is less than unguided wave. ë vgvp=v 2. ë As kx ® 0. vp ® ¥. Note, no violation of Special Relativity since energy is transmitted at vg. ë In the large k limit, behaviour approaches that of an unguided wave D Cut-off frequency ë No modes with real k for w<pv/b. This is the cutoff frequency. Below this, kx 2<0 and the wave is evanescent. 10 Waves 11

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

Evanescent waves D Below the cut-off frequency ë In the guide, ë below the cut-off frequency, ë kx 2 is negative, so ë The wave has the form: with a real. Oscillates with t ë Not oscillatory in the x-direction. ë An evanescent wave. 12 Waves 11

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