Surface Waves

Chris Linton

A (very loose) definitionA surface wave is a wave which propagates along the interface between two different media and which decays away from this interface

decay

decay

direction of propagation

Mathematical preliminaries

• linear theory (small oscillations)• time-harmonic motion

F(x,t) = Re[ f(x) e-iwt ]

w is the angular frequency ( /2w p is in Hz)• f(x) is complex – it describes both the amplitude

and phase of the wave• eikx represents a wave travelling in the x-direction

with wavelength l = 2 /p k

Water waves

u(x) = f(x)

2 f = 0

-w2f + gfz + (s/r)fzzz= 0gravity

surface tension

x

z

fluid velocity

Laplace’s equation

decay

z

x f = eikxekz

dispersion relation

w2 = gk + sk3/r

g ≅ 9.8 ms-1, water: r ≅ 1000 kgm-3, s ≅ 0.07 Nm-2

wavelength, l = 2p/k

speedc = w/k

1 ms-1

50 cm17 mm

try

Elastic waves

In an infinite elastic solid, two types of waves can propagate

u = uL + uT = f + ×y

longitudinal (P) waves, speed cL

transverse (S) waves, speed cT

cT < cL

In rock, cL 6 kms≅ -1, cT 3.5 kms≅ -1,

Rayleigh waves

z

xf = Aeikxekaz

y = (0,Beikxekbz,0)Navier’s equation

zero traction

decay u is in the (x,z)-plane

• Surface waves exist, with speed cR < cT (< cL)

• The quantity g = (cR/cT)2 satisfies the cubic equation

g3 - 8g2 + 8 (3-2 ) - 16(1- ) g L L = 0• When L = 1/3, we find that cR ≅ 0.9cT

• Non-dispersive (cR does not depend on w)

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

http://www.yorku.ca/esse/veo/earth/sub1-10.htm

Rayleighwave

Lovewave

http://web.ics.purdue.edu/~braile/edumod/waves/WaveDemo.htm

SAW devices

http://tfy.tkk.fi/optics/research/m1.php

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 deviceThey are used, e.g., in radar equipment, TVs and mobile phonesWorldwide, about 3 billion SAW devices are produced annually

Electromagnetic surface waves

x

y

z

e,m

e,m

E = Ê eilz, H = Ĥ eilz

Maxwell’s equations show that the field is determined from Êz and Ĥz.Both satisfy the Helmholtz equation

2u+(k2-l2)u=0

k2 = emw2/c2

Tangential components of E and H must be continuous on r = (x2+y2)1/2 = a

Require decay as r ∞ k’2 = e’m’w2/c2

Single mode optical fibres

Try Êz = A Jm(ar) eimq, a2 = k2-l2 B Km( a r) eimq, a2 = l2-k2

k2 < l2 < k2

Except when m = 1, there is a critical radius below which waves of a given frequency cannot propagate

The exception is often called the HE1,1 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

Edge waves

A continental shelf mode. From Cutchin & Smith, J. Phys. Oceanogr. (1973)

zx

a

Kf = fz

fn = 0decay

2 f = 0f = eilye-l(x cos a – z sin a)

K = l sin a

K = w2/g

rigid boundary

Stokes (1846)

Extended by Ursell (1952)

K = l sin (2n+1)a

(2n+1)a < p/2

dispersion relation

Array guided surface waves

decay

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)

McIver, CML & McIver (1998)

antisymmetric modes are also possibledet(dmn+Zmsn-m(b)) = 0

quasiperiodicityf(x+1,y) = eibf(x,y)

1D array in 2D

acoustic waves, rigid cylinders

a = 0.25, k = w/c = 2.5, b = 2.59

2 f +k2 f = 0

dispersion curves, symmetric modes

k

b

a = 0.125 a = 0.25

a = 0.375

0 < k < b ≤ p

Excitation of AGSWsThompson & CML (2007)

AGSWs on 2D lattices in 3D

s1

s2

quasiperiodicityRpq = ps1+qs2

f(r+Rpq) = eiRpq. b f(r) b is the Bloch vector

b can be restricted to the ‘Brillouin zone’ and we require |b| > k

det(dmn+Zmsn-m(b)) = 0

in plane out of plane

s1 = (1,0), s2 = (0.2,1.2), k = 2.8, a = 0.3, arg b = p/4, |b| = 2.807

Thompson & CML (2010)

Water waves over periodic arrayof horizontal cylinders

eily dependenceKf = fz K = w2/g

(2–l2) f = 0fn = 0 on rj=a decay

bd – ld dispersion curves

f/d=0.5, a/d=0.25, Kd=2,3,4,5,6,7energy propagates normal to these isofrequency

curves in the direction of increasing K

Transmitted energy over a finite array

Kd=4, f/d=0.5, a/d=0.25band gap for Kd=4 corresponds to ld in (2.808,3.017), or angle of incidence between 44.6 and 49.0 degrees

CML (2011)

41°

43°

45°

47°49°

50°

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