108
4 After the Lips: Acoustic Resonances and Radiation
Fig. 4.5 Sound radiation from the end of a cylindrical tube
Equation 4.4 relates the behaviour of the sound wave to the boundary condition
imposed by the solid wall of the confining cylinder. This boundary condition
changes suddenly at the open end of the tube: the wave is now free to propagate
in any direction in an effectively infinite space. Mathematical solution of the wave
equation shows that outside the tube the wavefronts are no longer plane, but are
close to being spherical, as illustrated in Fig. 4.5.
The nature of the sound waves radiated from different types of tube termination
will be discussed in detail in Sect. 4.6. Here we are interested in the effect of
the opening on the internal sound in the tube. An abrupt change in boundary
conditions is usually accompanied by a strong reflection of the wave. As discussed
in Sect. 2.2.5, most of the sound energy arriving at the open end is not radiated in
the spherical wave, but reflected back up the tube as a plane wave p − , described
mathematically as
p − = Be
j (ωt+kx) .
(4.14)
The forward and backward travelling waves exist simultaneously in the tube, and
at any give time t and place x, the total acoustic pressure is simply the sum of the
pressures in the two waves:
p(x, t) = p + (x, t) + p − (x, t) = Ae
j (ωt−kx)
+ Be
j (ωt+kx) .
(4.15)
We chose to start the clock measuring t in Fig. 4.4 when a crest of the pressure
wave was passing the point x = 0. This choice determined that the amplitude A of
the forward-going wave was a real positive number. The amplitude of the backward
going wave is determined by the boundary condition at x = L. Just outside the open
end, the acoustic pressure is very close to zero, since any rise or fall is immediately
compensated by air flow from the surrounding atmosphere. As a first approximation,
we assume that p(L, t) = 0 for all values of t. Equation 4.15 shows that this is true
if
Be
jkL
= −Ae
−jkL
4 After the Lips: Acoustic Resonances and Radiation
Fig. 4.5 Sound radiation from the end of a cylindrical tube
Equation 4.4 relates the behaviour of the sound wave to the boundary condition
imposed by the solid wall of the confining cylinder. This boundary condition
changes suddenly at the open end of the tube: the wave is now free to propagate
in any direction in an effectively infinite space. Mathematical solution of the wave
equation shows that outside the tube the wavefronts are no longer plane, but are
close to being spherical, as illustrated in Fig. 4.5.
The nature of the sound waves radiated from different types of tube termination
will be discussed in detail in Sect. 4.6. Here we are interested in the effect of
the opening on the internal sound in the tube. An abrupt change in boundary
conditions is usually accompanied by a strong reflection of the wave. As discussed
in Sect. 2.2.5, most of the sound energy arriving at the open end is not radiated in
the spherical wave, but reflected back up the tube as a plane wave p − , described
mathematically as
p − = Be
j (ωt+kx) .
(4.14)
The forward and backward travelling waves exist simultaneously in the tube, and
at any give time t and place x, the total acoustic pressure is simply the sum of the
pressures in the two waves:
p(x, t) = p + (x, t) + p − (x, t) = Ae
j (ωt−kx)
+ Be
j (ωt+kx) .
(4.15)
We chose to start the clock measuring t in Fig. 4.4 when a crest of the pressure
wave was passing the point x = 0. This choice determined that the amplitude A of
the forward-going wave was a real positive number. The amplitude of the backward
going wave is determined by the boundary condition at x = L. Just outside the open
end, the acoustic pressure is very close to zero, since any rise or fall is immediately
compensated by air flow from the surrounding atmosphere. As a first approximation,
we assume that p(L, t) = 0 for all values of t. Equation 4.15 shows that this is true
if
Be
jkL
= −Ae
−jkL