248
5 Blow That Horn: An Elementary Model of Brass Playing
F (X) =
dX
dt
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
dX(1)
dt
dX(2)
dt
dX(3)
dt
. . .
dX(2 + N)
dt
dX(2 + N + 1)
dt
. . .
dX(2 + N + N)
dt
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
dh
dt
d 2 h
dt 2
dp 1
dt
. . .
dp N
dt
d 2 p 1
dt 2
. . .
d 2 p N
dt 2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(5.31)
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
X(2)
−ω 2
l X(1) −
ω l
Q l
X(2) −
2+N
n=3 X(n)
μ
+ ω 2
l h 0 +
p m
μ
X(2 + N + 1)
. . .
X(2 + N + N)
−
ω 1
q 1
X(2 + N + 1) − ω 2
1 X(3) + Z 1
ω 1
q 1
du
dt
. . .
−
ω N
q N
X(2 + N + N) − ω 2
N X(2 + N) + Z N
ω N
q N
du
dt
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
(5.32)
with
du
dt
= wX(2)
2
ρ
p m −
2+N
n=3
X(n)
1/2
−
w
ρ
X(1)
2+N +N
n=2+N +1
X(n)
2
ρ
p m −
2+N
n=3
X(n)
−1/2
.