15. Musical Sound, Noise, Resonance and Beats

15.1 Musical Sound vs Noise

Sounds can broadly be classified into two categories based on the regularity of the vibrations that produce them. A musical sound (or musical note) is produced by regular, periodic vibrations of nearly constant frequency and amplitude, giving rise to a smooth, pleasant, and consistent sensation to the listener — examples include the sounds produced by musical instruments (a tuned violin string, a flute, a piano) and a well-trained singing voice. Noise, on the other hand, is produced by irregular, non-periodic (aperiodic) vibrations with rapidly and randomly varying frequency and amplitude, giving rise to a harsh, jarring, or unpleasant sensation — examples include the roar of traffic, the grinding of machinery, and random loud shouting. It is worth noting that this distinction is fundamentally about the regularity of the underlying vibration pattern, not simply about loudness — a musical sound can be very loud, and noise can, in principle, be relatively soft, though in everyday usage 'noise' also often connotes unwanted or excessively loud sound (see noise pollution, below).

15.2 Resonance

Every object capable of vibrating has one or more natural frequencies (also called its natural or characteristic frequency) — the specific frequency or frequencies at which it will vibrate most readily and with the least energy input if simply disturbed and left free to oscillate on its own, determined by its physical properties such as size, shape, mass, elasticity, and tension. Resonance is the phenomenon in which an object is made to vibrate with a dramatically increased amplitude when it is subjected to an external periodic force (or an external sound wave) whose frequency exactly matches (or very closely matches) the object's own natural frequency. When this frequency match occurs, energy is transferred from the driving force/wave into the object's vibration with maximum efficiency at each cycle, causing the amplitude of vibration to build up progressively, often to a surprisingly large and sometimes destructive extent, even from a comparatively weak driving force, provided the matching frequency is sustained over enough cycles.

Everyday and technical examples of resonance include: pushing a swing at exactly the same rhythm as its own natural back-and-forth frequency, causing it to swing higher and higher with each push even though each individual push applies only a small force; a singer capable of shattering a wine glass by singing a sustained note that exactly matches the glass's natural frequency of vibration; the tuning of a radio or television receiver, which works by electronically adjusting a circuit's natural (resonant) frequency to match the frequency of the desired broadcast signal, so that circuit responds strongly (resonates) to that particular station while effectively ignoring all others; and, as a cautionary and historically significant example, the collapse of certain bridges (most famously the Tacoma Narrows Bridge in 1940 in the United States) when wind-induced periodic forces happened to match the bridge structure's natural frequency of vibration, causing catastrophically large-amplitude oscillations that ultimately tore the structure apart — this is also why marching soldiers are conventionally instructed to break step (stop marching in unison) when crossing a bridge, to avoid the risk of their rhythmic, synchronised footsteps accidentally matching and exciting the bridge's natural frequency through resonance.

15.3 Beats

Beats are a periodic, regular rise and fall (a 'throbbing' or pulsing pattern) in the loudness of sound that is heard when two sound waves of slightly different, but very close, frequencies are made to superpose (overlap) with each other, such as when two musical notes that are nearly, but not quite, in tune are sounded together. At any given moment, the two overlapping waves may be in phase with each other (their compressions and rarefactions coincide, so the two waves reinforce each other and produce a moment of increased/maximum loudness) or out of phase with each other (the compression of one wave coincides with the rarefaction of the other, so the waves partially or wholly cancel each other out, producing a moment of decreased/minimum loudness, or near silence) — and because the two original frequencies are only slightly different from one another, the pattern of alternating reinforcement and cancellation repeats itself periodically at a slow, clearly perceptible rate, heard by the listener as a rhythmic throbbing or 'wa-wa-wa' pulsing in loudness. The number of such beats heard per second (the beat frequency) is exactly equal to the numerical difference between the two individual sound frequencies being superposed: beat frequency = |f₁ − f₂|. This phenomenon of beats is put to highly practical use by musicians in tuning instruments: two notes (for instance, from two guitar strings, or an instrument being tuned against a fixed reference such as a tuning fork or an electronic tuner) that are slightly out of tune with each other will produce audible beats when sounded together, and the musician adjusts the tension (and hence the frequency) of the instrument until the beats slow down and eventually disappear entirely, indicating that the two frequencies have become exactly equal and the instrument is now correctly in tune.