Biomedical Engineering Reference
In-Depth Information
shock (such as a clap); they can have a pitch (such as a canary song) or not
(such as the wind whispering through the trees). Sound waves can even seem
to be localized in space, as in the “hot spots” that occur when we sing in our
bathroom: sound appears and disappears according to our location.
What is a sound wave? It is the propagation of a pressure perturbation
(in much the same way as a push propagates along a line). Mathematically
speaking, a sound wave is a solution to the acoustic wave equation. By this
we mean a function p = p ( x, t ) satisfying (1.7). Every sound wave referred to
in the paragraph above can be described mathematically by an appropriate
solution to the acoustic wave equation. The buzz of a light tube or a note sung
by a canary, for instance, can be described by a traveling wave . What is the
mathematical representation of such a wave? Let us analyze a spatiotemporal
function of space and time of the following form:
p ( x, t )= p ( x
ct ) .
(1.8)
If we call the difference x − ct = u , then it is easy to see that taking the
time derivative of the function twice is equivalent to taking the space deriv-
ative twice and multiplying by c 2 . The reason is that ∂p/∂x = dp/du , while
∂p/∂t =
cdp/du . In other words, a function of the form (1.8) will satisfy the
equation (1.7). Interestingly enough, it represents a traveling disturbance. We
can visualize this in the following way: let us take a “picture” of the spatial
disturbances of the problem by computing p 0 = p ( x, 0). The picture will look
exactly like a picture taken at t = t , if we displace it a distance x = ct .
It is interesting to notice that just as a function of the form (1.8) satisfies
the wave equation, a function of the form p ( x, t )= p ( x + ct ) will also satisfy
it. In other words, waves traveling in both directions are possible results of
the physical processes described above. Maybe even more interestingly, since
the wave equation (1.8) is linear, a sum of solutions is a possible solution.
The spatiotemporal patterns resulting from adding such counterpropagating
traveling waves are very interesting, and can be used to describe phenomena
such as the “hot spots” in the bathroom. They are called “standing waves”
and will be discussed as we review some elements that are useful for their
description.
1.1.5 Detecting Sound
To detect sound, we need somehow to measure the pressure fluctuations. One
way to do this is to use a microphone, which is capable of converting pressure
fluctuations into voltages. Now we are able to analyze Fig. 1.2, which is a typ-
ical display of a record of a sound. The sound wave, that is, the propagation
of a pressure perturbation, reaches our microphone and moves a mechanical
part. This movement induces voltages in a circuit, which are recorded. In
Fig. 1.2, we have plotted the voltage measured (which is proportional to the
pressure of the sound wave in the vicinity of the microphone) as a function of
Search WWH ::




Custom Search