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Surface Waves Chris Linton

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A (very loose) definition A surface wave is a wave which propagates along the interface between two different media and which decays away from this interface decay direction of propagation

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Mathematical preliminaries linear theory (small oscillations) time-harmonic motion F(x,t) = Re[ f(x) e -i t ] is the angular frequency ( is in Hz) f(x) is complex – it describes both the amplitude and phase of the wave e i kx represents a wave travelling in the x-direction with wavelength = k linear theory (small oscillations) time-harmonic motion F(x,t) = Re[ f(x) e -i t ] is the angular frequency ( is in Hz) f(x) is complex – it describes both the amplitude and phase of the wave e i kx represents a wave travelling in the x-direction with wavelength = k

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Water waves u x = x 2 = - 2 + g z + ( ) zzz = 0 gravity surface tension x z fluid velocity Laplace’s equation decay

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z x = e ikx e kz dispersion relation 2 = gk + k 3 / g ≅ 9.8 ms -1, water: ≅ 1000 kgm -3, ≅ 0.07 Nm -2 wavelength, = 2 /k speed c = /k 1 ms cm 17 mm try

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Elastic waves In an infinite elastic solid, two types of waves can propagate u = u L + u T = × longitudinal (P) waves, speed c L transverse (S) waves, speed c T c T < c L In rock, c L ≅ 6 kms -1, c T ≅ 3.5 kms -1,

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Rayleigh waves z x = Ae ikx e k z = (0,Be ikx e k z,0) Navier’s equation zero traction decay u is in the (x,z)-plane Surface waves exist, with speed c R < c T (< c L ) The quantity = (c R /c T ) 2 satisfies the cubic equation = When = 1/3, we find that c R ≅ 0.9c T Non-dispersive (c R does not depend on ) Surface waves exist, with speed c R < c T (< c L ) The quantity = (c R /c T ) 2 satisfies the cubic equation = When = 1/3, we find that c R ≅ 0.9c T Non-dispersive (c R does not depend on )

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Earthquakes Lord Rayleigh (1885) “It is not improbable that the surface waves here investigated play an important part in earthquakes” Lord Rayleigh (1885) “It is not improbable that the surface waves here investigated play an important part in earthquakes” Rayleigh wave Love wave

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SAW devices In the 1960s it was realised that Surface Acoustic Waves (Rayleigh waves) could be put to good use in electronics There are many types of SAW device They are used, e.g., in radar equipment, TVs and mobile phones Worldwide, about 3 billion SAW devices are produced annually

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Electromagnetic surface waves x y z ,, ,, ,, ,, E = Ê e i l z, H = Ĥ e i l z Maxwell’s equations show that the field is determined from Ê z and Ĥ z. Both satisfy the Helmholtz equation 2 u+(k 2 - l 2 )u=0 k 2 = c Tangential components of E and H must be continuous on r = (x 2 +y 2 ) 1/2 = a Require decay as r ∞ k’ 2 = ’ ’ c

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Single mode optical fibres Try Ê z = A J m ( r) e im 2 = k 2 - l 2 B K m ( r) e im 2 = l 2 -k 2 k 2 < l 2 < k 2 Except when m = 1, there is a critical radius below which waves of a given frequency cannot propagate The exception is often called the HE 1,1 mode and single mode optical fibres can be fabricated with diameters of the order of a few microns m = 0,1,2,… Theory 1910, practical importance 1930s & 1940s, realisation 1960s

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Edge waves A continental shelf mode. From Cutchin & Smith, J. Phys. Oceanogr. (1973) z x K = z n = 0 decay 2 = = e i l y e - l (x cos – z sin ) K = l sin K = 2 /g rigid boundary Stokes (1846) Extended by Ursell (1952) K = l sin (2n+1) (2n+1) < /2 dispersion relation

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Array guided surface waves decay 1D array in 2D 1D array in 3D 2D array in 3D waves exist due to the periodic nature of the geometry Barlow & Karbowiak (1954)

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McIver, CML & McIver (1998) antisymmetric modes are also possible det( mn +Z m n-m ( )) = 0 quasiperiodicity (x+1,y) = e i (x,y) quasiperiodicity (x+1,y) = e i (x,y) 1D array in 2D acoustic waves, rigid cylinders a = 0.25, k = /c = 2.5, = 2.59 2 +k 2 = 0

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dispersion curves, symmetric modes k a = a = 0.25 a = < k < ≤

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Excitation of AGSWs Thompson & CML (2007)

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AGSWs on 2D lattices in 3D s1s1 s2s2 quasiperiodicity R pq = ps 1 +qs 2 (r+R pq ) = e iR pq. (r) is the Bloch vector quasiperiodicity R pq = ps 1 +qs 2 (r+R pq ) = e iR pq. (r) is the Bloch vector can be restricted to the ‘Brillouin zone’ and we require | | > k det( mn +Z m n-m ( )) = 0

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in planeout of plane s 1 = (1,0), s 2 = (0.2,1.2), k = 2.8, a = 0.3, arg = /4, | | = Thompson & CML (2010)

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Water waves over periodic array of horizontal cylinders e i l y dependence K = z K = 2 /g ( 2 – l 2 ) = n = 0 on r j =a decay

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d – l d dispersion curves f/d=0.5, a/d=0.25, Kd=2,3,4,5,6,7 energy propagates normal to these isofrequency curves in the direction of increasing K

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Transmitted energy over a finite array Kd=4, f/d=0.5, a/d=0.25 band gap for Kd=4 corresponds to l d in (2.808,3.017), or angle of incidence between 44.6 and 49.0 degrees CML (2011) 41° 43° 45° 47° 49° 50°

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Summary Surface waves occur in many physical settings Mathematical techniques that can be used to analyse surface waves are often applicabe in many of these different contexts There is often a long time between the theoretical understanding of a particular phenomenon and any practical use for it The study of array guided surface waves is in its infancy Surface waves occur in many physical settings Mathematical techniques that can be used to analyse surface waves are often applicabe in many of these different contexts There is often a long time between the theoretical understanding of a particular phenomenon and any practical use for it The study of array guided surface waves is in its infancy

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