Page 291 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
different types of waves that we can effectively guide. Some of these sur-
face waves have special names:
• Beams: Flexural waves. (Daniel Bernoulli, 08/02/1700–17/03/1782,
Leonard Euler, 15/04/1707–1783, and Stephen P. Timoshenko, 22/12/
1878–29/05/1972).
• Homogeneous half-space: Rayleigh waves (John William Strutt Lord
Rayleigh, 12/11/1842–30/06/1919).

• Plates: Lamb waves (Horace Lamb, 29/11/1849–4/11/1934).
• Thin layer on an elastic half-space: Love waves (Augustus Edward
Hough Love, 17/04/1863–5/06/1940).

The basic analysis technique is to first make a unique decomposition of
the displacement field into two parts u = u′ + u″ . One component is
divergence-free and the other is rotation-free. Because of this separation,
φ
we can assume the existence of a scalar potential and a vector potential
ψψ ψ ψ (as we did for the electrodynamic potentials ψ and A ), and hence
obtain the unique decomposition
u = ∇× ψψ ψ ψ + ∇ φ (7.113)

Elasticity for The analysis is greatly simplified for the Lamb and Raleigh waves if we
Raleigh and choose the direction of propagation parallel to a coordinate axis, here the
Lamb Waves
u
x-axis. In addition, we assume no dependence of on the y-direction
(for the plate this means a plane strain assumption), so that
,,
ψψ ψ ψ = ( 0 ψ 0) and hence
------ 0 ------- +
------- +
------
u = ( u 0 u,, ) =  – ∂ψ ∂φ ,, ∂ψ ∂φ  (7.114)
1 3  ∂z ∂x ∂x ∂z 
From elasticity, we recall that the “small” strain is defined component-
wise as

1 ∂u i ∂u j

ε = --- -------- + -------- (7.115)
ij  ∂x 
2 ∂x j i

288 Semiconductors for Micro and Nanosystem Technology