4.7 Going Further: Calculating Input Impedance
201
Fig. 4.90 Approximation of
a flaring bell as series of
cylindrical sections
For the backward travelling wave, the acoustic particle velocity (measured in the +x
direction) changes sign, so
v − (x, t) = −
B
ρc
e
j (ωt+kx)
= −
1
ρc
p − (x, t).
(4.89)
The total acoustic particle velocity at x is therefore
v(x, t) =
1
ρc
(p + (x, t) − p − (x, t)) =
1
ρc
Ae
j (ωt−kx)
− Be
j (ωt+kx)
.
(4.90)
Recalling that the volume velocity u(x, t) = Sv(x, t), where S is the cross-sectional
area of the cylinder, and the characteristic impedance of the cylinder is Z c = ρc/S
(Eq. 4.31), the total volume velocity at x is
u(x, t) =
1
Z c
Ae
j (ωt−kx)
− Be
j (ωt+kx)
.
(4.91)
We consider the cylindrical section of length L starting at x 1 and ending at x 2
and derive expressions relating the pressure and volume velocity at the entrance to
the corresponding quantities at the exit. To simplify the notation, the common factor
exp(j ωt) is omitted. From Eq. 4.15,
p(x 1 ) = Ae
−jk(x 2 −L)
+ Be
jk(x 2 −L)
= Ae
−jkx 2 (cos(kL) + j sin(kL)) + Be
jkx 2 (cos(kL) − j sin(kL))