Biomedical Engineering Reference
In-Depth Information
relative
amplitude
F
3F
5F
frequency
Fig. 2.6. Response function (or gain) for a tube with one end open and the other
closed. The response function is the way the tube responds when excited with
a sound wave. The peaks are called the resonances of the tube. In a real tube,
the peaks are not ideally narrow like these but have a width determined by the
energy losses. The frequencies corresponding to the maxima of the resonances are
the natural frequencies of the tube. A sound wave propagating along the tube will
resonate or increase its amplitude if it has a frequency very close to a natural
frequency of the tube. Otherwise, it will be attenuated
2.2.5 Standing Waves
We have discussed the behavior of the density (or pressure) fluctuations in the
vicinity of the exciting membrane. It is time to touch upon a most interesting
issue, related to the spatial distribution of the density along the tube, when
the membrane is vibrating at one of the natural frequencies.
Let us pay attention to the signal generated by the membrane at the
first natural frequency F 1 = c/ (4 L ). According to the arguments presented
in Sect. 2.2.3, the reflected wave will return to the vicinity of the exciting
membrane in such a way that it contributes constructively to the amplitude
of the total signal. We say that the reflected wave and the injected signal are
in phase, and that the total amplitude of the pressure fluctuations will be
maximum. This can be mathematically described as a pattern p e = p e ( x, t )
as follows:
p e = p e 0 cos( kx
ωt )+ p e 0 cos( kx + ωt ) ,
(2.17)
where k =2 π/ (4 L )and ω =2 πf . Making use of the trigonometric identity
cos( α
±
β )=cos α cos β
sin α sin β , we can rewrite (2.17) as
p e =2 p e 0 cos( kx )cos( ωt ) ,
(2.18)
which is called a standing wave. Functions such as this, with a factorized
space and time dependence, are also solutions to the wave equation (1.7)
(see Fig. 2.3b for snapshots of a generic standing wave). This result can be
interpreted as a temporal oscillation that has a position-dependent amplitude
2 P e 0 cos( kx ). Notice that at x = 0 the amplitude of the oscillation is always
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