Digital Signal Processing Reference
In-Depth Information
A (mechanical) cavity resonator is, for example, any type of flute, organ pipe or the
resonant cavity of a guitar.
Depending on the shape and size of the air volume, particular frequencies or frequency
ranges are amplified in a resonator, others are attenuated. The frequency ranges are
amplified for which standing waves can be produced in the cavity.
Transitions are fluid because a flute can be defined as an oscillator or a resonator. In either
case it is a vibrating system.
Let us first concentrate on a one-dimensional vibrating system, a string of a guitar. At
either end it is firmly fixed and can thus not be deflected there. When the string is plucked
it always produces the same tone or at least the same pitch. Why is that?
A fixed vibrating string is a one-dimensional oscillator/resonator. When it is plucked -
which is the supply of energy - it produces free-running oscillations in its characteristic
frequencies (natural frequencies; free-running oscillations).
If the string were stimulated in a periodic sinusoidal way with varying frequencies it
would oscillate in a controlled way. In that case the frequencies leading to an extreme
deflection - in the form of standing sinusoidal waves - are called resonance frequencies.
The natural frequencies are identical with the resonance frequencies but are theoretically
slightly lower because the attenuation leads to a delay in the oscillation process.
Plucking the string means stimulating it in all frequencies because a one-off short pulse
contains basically all the frequencies (see Illustration 47). But all the sinusoidal waves
running up and down the string - reflected at the ends of the string - delete each other,
except those which are amplified by interference and which produce standing waves with
nodes and antinodes. The nodes are those points of the string which are permanently
immobile, i.e. which are not deflected. These natural frequencies are always integral
multiples of a base frequency, i.e. a string oscillates in a near-periodic way and thus
produces a harmonic tone. The wavelength can be calculated quite easily by taking the
length of the string L.
Definition
:
The wavelength
of a sinusoidal wave is the distance covered by the wave in the
period length T. For the velocity of the wave c the following is true:
λ
c = distance / time =
λ
/ T and as f = 1 / T it follows
c =
λ
<
f
Example:
The wavelength
λ
of a soundwave (c = 336 m/s) of 440Hz is 0.74m.
For the standing wave on the string of an instrument only sinusoidal waves for which the
length of the string is an integer multiple of
/2 are possible. This is to be seen clearly in
Illustration 74. All the other sinusoidal waves either peter out alongside the string or
eliminate each other.
λ
