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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