Biomedical Engineering Reference
In-Depth Information
phenomena, from electromagnetic waves [Feynman et al. 1970] to waves of
calcium concentration inside a living cell [Keener and Sneyd 1998].
Traveling waves behave interestingly when they reach “boundaries”.
Imagine a uniform tube, open at one end and closed at the other by a mem-
brane capable of vibrating. The motion of this membrane will push the air
in its surroundings, inducing fluctuations of density and pressure that can
propagate along the tube. This propagation can be described by a function
p
e
=
p
e
(
kx
ωt
), as discussed above. Now, what happens as the perturba-
tion of pressure arrives at the open end of the tube? This is very interesting:
an important fraction of this pressure perturbation is reflected,
inverted
,so
that the sum of the incident pressure perturbation
p
e
and the perturbation
induced by the reflected wave
p
r
give a negligible total pressure perturbation
at the open end of the tube:
p
e
(
x
=
L, t
)+
p
r
(
x
=
L, t
)
−
0. Why? Because
the atmosphere imposes its own pressure on the daring attempt of the tube
to try to change the pressure with its tiny pressure fluctuations. This is a
mechanism similar to that observed when we generate a wave on a string
with one end attached to a wall (see Fig. 2.4). The wave propagates and is
reflected, inverted. The sum of the displacements (incident and reflected) at
the fixed end must be zero, for the end of the string is rigidly attached to the
wall. In the case of the tube, the pressure at the open end is “tied” to the
atmospheric pressure. Notice that the fact that the pressure fluctuation is
zero does not mean that the interior of the tube is isolated from the exterior.
The displacement
D
, as seen in Chap. 1, satisfies
∂D/∂x
∼
p
and therefore
it oscillates with the maximum possible amplitude at the open end. These
fluctuations of the displacement are responsible for sound emission from the
endofthetube.
∼−
Fig. 2.4.
A wave on a string attached to a wall. If you flap the string to create a
propagating wave, the wave travels down the string until it is reflected at the end
attached to the wall, and becomes inverted. A pressure sound wave propagating
along a tube will be reflected and inverted at an
open
end of the tube in an analogous
way
