WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1628 J. S. JENSEN
Equation (A10) contains terms that depend on b ′ i . By differentiating Equation (26) b′ i can be
expressed in terms of u ′ i
⎛
b
⎜
⎝
′ 1
b ′ ⎞ ⎡
(u
⎟ ⎢
2 ⎟ ⎢
⎟ ⎢
⎟ =−⎢
⎟ ⎢
. ⎠ ⎢
⎣
+
1 )′
(u +
2 )′
⎤ ⎡
u
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ uN+1 − ⎢
. ⎥ ⎢
⎦ ⎣
+
1
u +
⎤
⎥
2 ⎥ u
. ⎥
⎦
′ N+1
(A13)
(u + N )′
b ′ N
(u ∗ 1 )′
(u + N )′
An expression for (u + i )′ is now needed. Differentiating Equation (27) gives
⎡
(u
⎢
P ⎢
⎣
+
1 )′
(u +
2 )′
⎤ ⎡
(u
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ = ⎢
. ⎥ ⎢
⎦ ⎣
∗ N )′
(u ∗ N−1 )′
⎤ ⎡
(u
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ − ⎢
⎥ ⎢
. ⎦ ⎣
∗ N )′
(u ∗ N−1 )′
⎤
⎡
u
⎥
⎢
⎥
⎢
⎥
⎢
⎥ [uN uN−1 ... u1] ⎢
⎥
⎢
. ⎦
⎢
⎣
+
1
u +
⎤
⎥
2 ⎥
. ⎥
⎦
⎡
⎢
− ⎢
⎣
u ∗ N
u ∗ N−1
.
u ∗ 1
where it is recalled that P is defined as
⎡
⎢
P = ⎢
⎣ .
(u ∗ 1 )′
u + N
⎤
⎥ [u
⎥
⎦
′ N u′ N−1 ... u′ 1 ]
⎡
u
⎢
⎣
+
1
u +
⎤
⎥
2 ⎥
. ⎥
⎦
u ∗ N
u ∗ N−1
u ∗ 1
u + N
u + N
(A14)
⎤
⎥ [uN uN−1 ··· u1] (A15)
⎥
⎦
and () ∗ is the conjugate transpose vector.
Thus, if both sides of Equation (A14) is multiplied by Q = P−1 an expression for (u + i )′ is
obtained which can be inserted into Equation (A13) to find b ′ i . This is then used to build an
expression for ũib ′ i
ũib ′ i =−ũiu + i u′ N+1 −
N
Q j,N+1−iũj ˜h
j=1
T ū ′ i + u+
N+1−i uN+1
N
ũ j
j=1
˜Q T
j u ′ i (A16)
in which
h T = u T N
N+1 −
u
k=1
+ k uN+1u T N+1−k
(A17)
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme