Page 299 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
lie in the plane of the surface. The derivatives w.r.t.
where x
and k
p
p
time and space coordinates are
2
2
∂ u′ 2 ∂ u″ 2
---------- = – ω u′ ----------- = – ω u″ (7.132a)
2
2
∂t ∂t
2
2
∂ u′ 2 ∂ u″ 2
---------- = k – u′ ----------- = k – u″ (7.132b)
2 x 2 x
∂x ∂x
2
2
∂ u′ 2 ∂ u″ 2
---------- = k – u′ ----------- = k – u″ (7.132c)
y
y
2
2
∂y ∂y
2
2
∂ u′ ∂Fz() i k p x p –( ⋅ ωt) ∂ u″ ∂Gz() i k p x p –( ⋅ ωt)
---------- = --------------e ----------- = ---------------e (7.132d)
2
2
∂z ∂z ∂z ∂z
2
2
∂ u′ ∂ u″
------------ = – k k u′ ------------ = – k k u″ (7.132e)
x y
x y
∂x∂y ∂x∂y
2
2
∂ u′ ∂Fz() ∂ u″ ∂Gz()
------------ = ik --------------u′ ------------ = k ---------------u″ (7.132f)
x
x
∂x∂z ∂z ∂x∂z ∂z
2
2
∂ u′ ∂Fz() ∂ u″ ∂Gz()
------------ = i--------------k u′ ------------ = i---------------k u″ (7.132g)
y
y
∂z∂y ∂z ∂z∂y ∂z
since we assume no dependence of the motion on the y-direction.
Lamb Waves Plates are idealizations used when a geometrical object is flat and very
in Mindlin much thinner than its other dimensions. In this case we can conveniently
Plates
describe the geometry using two coordinates only, say x and y. For the
following formulation we closely follow [7.5]. We define the averaged
plate stresses (or bending moments and shear forces) by
t
---
{ M , M , M } = 2 t ∫ σzz (7.133a)
d
xx
yy
xy
– ---
2
t
---
{ Q , Q } = 2 t ∫ σ z (7.133b)
d
xz
yz
– ---
2
296 Semiconductors for Micro and Nanosystem Technology