Page 298 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Inhomogeneities
and hence (7.127) becomes
4
2
( EIk ) (– ρ ω ) +   I + EIρ m 2 2 Iρ m 2 4 0 (7.129)
-----------ω =
------------- k ω +

m
m
k
k
T
T
4
⁄
We next divide throughout by EIk and make use of c = ω k and the
⁄
bulk wave speed c = E ρ . Also note that I ⁄ I = ρ and set
m
B
s = k ⁄ ρ . This yields
T
 c 2   1 1  1
 2
 4
1 –  --------------- +  ----- + ---- c +  ---------- c = 0 (7.130)
2
2 2
2
2
 c i k  c 2 s c s
B g B B
We are interested in the real roots of this equation, which yield the dis-
persion curves ck() of the Timoshenko beam.
Raleigh Rayleigh waves are harmonic in the plane of the surface of the solid, and
Waves decay exponentially into the depth of the material. A convenient ansatz
for the displacement (see Figure 7.16) is
(
⋅
i k p x p – ωt)
u′ = Fz()e (7.131a)
(
⋅
i k p x p – ωt)
u″ = Gz()e (7.131b)
z
Figure 7.16. A surface acoustic
(
– ξz i ωt – kx)
Raleigh wave is represented by a e e
harmonic amplitude function that
decays exponentially along the z-
direction away from the surface
into the solid. The harmonic part
of the amplitude strength is plotted
as a surface in the vertical direc-
x
tion over the xz-plane.

Semiconductors for Micro and Nanosystem Technology 295