Sound diffusion systems - Resonant frequency of a loudspeaker

When an elastic system undergoes an oscillating stimulus, it reacts differently depending on the frequency content of the stimulations. In particular, an elastic system will begin to oscillate at the same frequency as the stimulation once the latter is similar his resonance frequency. Every elastic system has in fact its own resonance frequency which can be calculated by using mathematical formulae which describe the quantities involved in the system itself.

Let's now move on from theory to practice, and see what happens. Let's imagine that our elastic system is a loudspeaker (so, a system composed of different parts: membrane, coil etc.), which will have its own resonance-frequency which for our example's sake we will fix at 40Hz.

By applying a sinusoidal electric signal to the loudspeaker and varying the signal's frequency, we will see that until the signal's frequency is distant from that of the loudspeaker, the loudspeaker's membrane will not be stimulated (or perhaps only minimally). When we reach a frequency of about 40Hz the membrane also begins to oscillate at this rate, and we will begin to hear a sound coming out of the loudspeaker which corresponds to the frequency of the applied electrical signal.

The following diagram shows the amplitude of the oscillation in question, stimulated by a signal whose frequency we shall vary:

Stimulation of an elastic system

We can see how the amplitude of the oscillations is at its highest in proximity of the resonance-frequency, whereas elsewhere it is almost zero. The figure also shows the phase-diagram of this elastic system, which highlights how frequencies greater than the resonance-frequency are subject to phase inversion (a phase displacement of 180 degrees implies a polarity inversion, in other words, a phase inversion). Naturally this situation is highly undesired for a loudspeaker which mustn't introduce input-signal variations, and even more so a phase inversion in the frequency-band it has to reproduce. Indeed, the phase-diagram of a loudspeaker never has a rate as that shown on the figure, which however has been used as an example to highlight once again the problems concerning the phase's rate- problems which are easy to overlook.