4.1 Internal Sounds in Brass Instruments
109
or
B = −Ae
−2jkL
= −A(cos 2kL − j sin 2kL).
(4.16)
When a wave of unit pressure amplitude and zero phase travels down the tube,
the ratio
R(ω) = B/A
(4.17)
is a complex number describing the magnitude and phase of the reflected wave
which returns to the input. This ratio is called the reflection coefficient of the tube.
R(ω) is written as a function of angular frequency because for most types of tube,
including the flaring tubes characteristic of brass instruments, the magnitude and
phase of the reflected wave depend on the frequency of the wave. For the openended cylindrical tube described above, however,
R(ω) = −e
−2jkL ,
(4.18)
and its magnitude is independent of frequency. The use of the reflection coefficient
in wind instrument modelling is discussed further in Sect. 4.1.5.
It is evident from Eq. 4.16 that B is in general a complex number. It is purely real
if 2kL = nπ , where n is an integer, since in these cases, sin 2kL = 0. If n is an even
integer, cos 2kL = 1, and B = −A. This means that at x = 0, the total pressure is
p(0, t) = p + (0, t) + p − (0, t) = (A + B)e
jωt
= 0.
(4.19)
If n is an odd integer, cos 2kL = −1, and B = A. The total pressure at x = 0 is
then
p(0, t) = (A + B)e
jωt
= 2Ae
jωt .
(4.20)
For the tube illustrated in Fig. 4.4, 2kL = 15π , so the total pressure at the input of
the tube is described by Eq. 4.20.
4.1.3 Standing Waves
The addition of two travelling waves going in opposite directions gives rise to a
different type of wave, described as a standing wave. The reason for this description
is clear when we consider the behaviour of crests and troughs in the standing wave
setup by the forward and backward travelling waves discussed in Sect. 4.1.2. Figure
4.6 illustrates how the waves add together at four different stages in one cycle of
oscillation. Forward travelling waves are shown by dashed red lines, backward
travelling waves by dotted blue lines, and the standing wave sums by solid green
lines.
109
or
B = −Ae
−2jkL
= −A(cos 2kL − j sin 2kL).
(4.16)
When a wave of unit pressure amplitude and zero phase travels down the tube,
the ratio
R(ω) = B/A
(4.17)
is a complex number describing the magnitude and phase of the reflected wave
which returns to the input. This ratio is called the reflection coefficient of the tube.
R(ω) is written as a function of angular frequency because for most types of tube,
including the flaring tubes characteristic of brass instruments, the magnitude and
phase of the reflected wave depend on the frequency of the wave. For the openended cylindrical tube described above, however,
R(ω) = −e
−2jkL ,
(4.18)
and its magnitude is independent of frequency. The use of the reflection coefficient
in wind instrument modelling is discussed further in Sect. 4.1.5.
It is evident from Eq. 4.16 that B is in general a complex number. It is purely real
if 2kL = nπ , where n is an integer, since in these cases, sin 2kL = 0. If n is an even
integer, cos 2kL = 1, and B = −A. This means that at x = 0, the total pressure is
p(0, t) = p + (0, t) + p − (0, t) = (A + B)e
jωt
= 0.
(4.19)
If n is an odd integer, cos 2kL = −1, and B = A. The total pressure at x = 0 is
then
p(0, t) = (A + B)e
jωt
= 2Ae
jωt .
(4.20)
For the tube illustrated in Fig. 4.4, 2kL = 15π , so the total pressure at the input of
the tube is described by Eq. 4.20.
4.1.3 Standing Waves
The addition of two travelling waves going in opposite directions gives rise to a
different type of wave, described as a standing wave. The reason for this description
is clear when we consider the behaviour of crests and troughs in the standing wave
setup by the forward and backward travelling waves discussed in Sect. 4.1.2. Figure
4.6 illustrates how the waves add together at four different stages in one cycle of
oscillation. Forward travelling waves are shown by dashed red lines, backward
travelling waves by dotted blue lines, and the standing wave sums by solid green
lines.