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Surface Waves
Chris Linton
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
decay
Mathematical preliminaries
• linear theory (small oscillations)
• time-harmonic motion
F(x,t) = Re[ f(x) e-iwt ]
w is the angular frequency (w/2p 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 = 2p/k
Water waves
fluid velocity
u(x) = f(x)
gravity
-w2f + gfz + (s/r)fzzz= 0
surface tension
z
x
Laplace’s equation
2f = 0
decay
z
try
x
f = eikxekz
dispersion relation
w2 = gk + sk3/r
g ≅ 9.8 ms-1, water: r ≅ 1000 kgm-3, s ≅ 0.07 Nm-2
1 ms-1
1.0
0.8
speed
c = w/k
0.6
0.4
0.2
0.0
17 mm
0.1
0.2
0.3
wavelength, l = 2p/k
0.4
0.5
50 cm
Elastic waves
In an infinite elastic solid, two types of
waves can propagate
u = uL + uT = f + ×y
longitudinal (P) waves, speed cL
cT < cL
transverse (S) waves, speed cT
In rock, cL ≅ 6 kms-1, cT ≅ 3.5 kms-1,L =
cT2
cL2
@ 1/ 3
Rayleigh waves
zero traction
f = Aeikxekaz
z
x
y = (0,Beikxekbz,0)
Navier’s equation
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 + 8g(3-2L) - 16(1-L) = 0
• When L = 1/3, we find that cR ≅ 0.9cT
• Non-dispersive (cR does not depend on w)
Earthquakes
Lord Rayleigh (1885)
“It is not improbable that the surface waves here
investigated play an important part in earthquakes”
Rayleigh
wave
Love
wave
http://www.yorku.ca/esse/veo/earth/sub1-10.htm
http://web.ics.purdue.edu/~braile/edumod/waves/WaveDemo.htm
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
http://tfy.tkk.fi/optics/research/m1.php
Electromagnetic surface waves
e,m
x
E = Ê eilz, H = Ĥ eilz
z
e,m
y
Maxwell’s equations show
that the field is determined
from Êz and Ĥz.
Both satisfy the Helmholtz
equation
Tangential components of E
and H must be continuous on
r = (x2+y2)1/2 = a
2u+(k2-l2)u=0
Require decay as r  ∞
k’2 = e’m’w2/c2
k2 = emw2/c2
Single mode optical fibres
Try Êz =
 B K (a r) e
A Jm(ar) eimq, a2 = k2-l2
m
imq,
a2 = l2-k2
k2 < l2 < k2
m = 0,1,2,…
Except when m = 1, there is a critical radius below
which waves of a given frequency cannot propagate
The exception is often called
the HE1,1 mode and single
mode optical fibres can be
fabricated with diameters of
the order of a few microns
Theory 1910, practical importance 1930s & 1940s,
realisation 1960s
Edge waves
Kf = fz
z
K = w2/g
x
a
2f = 0
fn = 0
rigid boundary
Stokes (1846)
f = eilye-l(x cos a – z sin
a)
dispersion relation K = l sin a
decay
Extended by Ursell (1952)
K = l sin (2n+1)a
(2n+1)a < p/2
A continental shelf mode. From Cutchin
& Smith, J. Phys. Oceanogr. (1973)
Array guided surface waves
decay
1D array in 2D
1D array in 3D
2D array in 3D
decay
waves exist due to
the periodic nature
of the geometry
Barlow & Karbowiak (1954)
6
McIver, CML & McIver (1998)
1D array in 2D
a = 0.25, k = w/c = 2.5, b = 2.59
4
acoustic waves,
rigid cylinders
2
2f +k2f = 0
0
quasiperiodicity
f(x+1,y) = eibf(x,y)
-2
antisymmetric
modes are
also possible
-4
det(dmn+Zmsn-m(b)) = 0
-6
-6
-4
-2
0
2
4
6
dispersion curves, symmetric modes
3.2
a = 0.125
2.8
a = 0.25
2.4
a = 0.375
2.0
k
1.6
1.2
0.8
0<k<b≤p
0.4
0.0
0.0
0.4
0.8
1.2
1.6
b
2.0
2.4
2.8
3.2
Excitation of AGSWs
Thompson & CML (2007)
AGSWs on 2D lattices in 3D
quasiperiodicity
Rpq = ps1+qs2
f(r+Rpq) = eiRpq.b f(r)
b is the Bloch vector
s2
s1
det(dmn+Zmsn-m(b)) = 0
b can be restricted to
the ‘Brillouin zone’ and
we require |b| > k
s1 = (1,0), s2 = (0.2,1.2), k = 2.8, a = 0.3, arg b = p/4, |b| = 2.807
Thompson & CML (2010)
2
2
in plane
out of plane
Water waves over periodic array
of horizontal cylinders
Kf = fz
K = w2/g
rj=a
z
x
a
fn = 0 on
eily dependence
qj
x=jd
z=-f
rj
d
(2–l2)f
=0
decay
bd – ld dispersion curves
3.0
2.5
2.0
bd
1.5
1.0
0.5
0.0
0
1
2
3
4
5
6
ld
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
7
Transmitted energy over a finite array
41°
1.0
0.8
43°
0.6
|TM|2
50°
41
0.4
43
45°
45
CML (2011)
47
0.2
49°
49
47°
50
0.0
0
10
20
30
40
50
60
M
Kd=4, f/d=0.5, a/d=0.25
band gap for Kd=4 corresponds to ld in (2.808,3.017),
or angle of incidence between 44.6 and 49.0 degrees
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