Resonance

RESONANCE. If any body (e.g., a pendulum, string, membrane, tuning fork, air column, etc.) or, in general, some system can perform independent oscillations with a period T (or with a frequency f = 1/T oscillations per second), then in the case of the action upon it of external oscillations with a period T₁ (or with a frequency f₁ = 1/T₁), two oscillations arise in it with periods T and T₁ (frequencies f and f₁). The first of these is called the natural oscillation of the system, the second—the forced oscillation. In the case of the equality of the periods T and T₁, the oscillations of the system become most intense, and this case bears the name of resonance of oscillations. Thus, for example, the oscillations of a tuning fork excite intense oscillations of the air column enclosed in the box upon which the tuning fork is mounted, if the natural frequency of the air mass enclosed in the box coincides with the frequency of the tuning fork; likewise, a string begins to oscillate and produces a sound audible to the ear under the influence of even weak acoustic air oscillations reaching it, if the frequency of the incoming oscillations is identical to the frequency of the natural oscillations of the string. Quite similarly, the concept of resonance is applied not only to a mechanical system, but to any system in which oscillations can arise, e.g., to an electrical oscillatory circuit, i.e., a circuit in which electrical oscillations (rapidly alternating currents) can exist. Electrical oscillations existing in one such circuit excite the most intense oscillations in a second circuit under the condition of equality of the natural periods of oscillation of the circuits, i.e., when the circuits are tuned to resonance. A special case is represented by electromechanical resonance, when, under the influence of electrical oscillations of a certain frequency, mechanical oscillations of the same frequency arise, or vice versa. This case occurs when using piezoelectric crystals, in particular quartz (for creating ultrasonic oscillations and in some radiotechnical circuits); under the influence of an alternating voltage applied to it, the quartz periodically compresses and expands.

The natural frequency of its mechanical oscillations is determined by its geometric dimensions, and the mechanical oscillations prove to be intense only at a frequency of the applied alternating voltage equal to the natural frequency of the mechanical oscillations of the crystal. The ability of a body to resonate not only at one specific frequency but also at adjacent frequencies can be characterized by the so-called resonance curve (see figure). Plotted on the horizontal axis here are the frequencies of the external oscillations acting on the system under consideration, and on the vertical axis—the amplitudes of the oscillations arising in the system; the natural frequency of the system is marked f_r. Curve A represents the case of "sharp resonance," in which the system resonates only at frequencies adjacent to its own; curve B relates to the case of "blunt" tuning. In the latter case, the system also responds to frequencies sufficiently distant from its natural frequency. The sharpness of resonance depends on the damping of oscillations in the given system. In the case where, in a system left to itself, the excited oscillations quickly cease, we say that the system possesses high damping; the less the damping in the system, the longer the oscillations are preserved in it. The less the damping, the sharper the resonance curve, and vice versa. An example of a system with low damping can be a tuning fork, which resonates only in a very narrow range of frequencies; an example of strong damping is the soundboard of a grand piano, which resonates at all frequencies.

In many cases of practical acoustics, the phenomenon of resonance proves to be a significant hindrance. Such is, for example, the resonance of microphone and telephone membranes, thanks to which certain sound pitches are amplified, resonance in loudspeaker horns, etc. According to the theory of Helmholtz, the perception of sounds of various pitches by the ear is based on the phenomenon of resonance. In this case, the resonating system consists of the fibers of the basilar membrane (membrana basilaris), stretched along the course of the cochlea. Each of the fibers of this membrane is a resonator tuned to a specific frequency. Air oscillations, transmitted from the outer ear to the cochlea, cause those fibers that are in resonance with the incoming sound waves to oscillate (along with these fibers, adjacent fibers also begin to oscillate, but their oscillations prove to be much weaker).