Acoustic Resonance

Acoustic resonance refers to the amplification of sound waves within a physical system when the frequency of an external excitation matches one of the system’s natural frequencies of vibration. When resonance occurs, the system absorbs energy efficiently and vibrates with increased amplitude, a principle observable in a wide range of natural and engineered structures. Acoustic resonance extends beyond the audible spectrum, as acoustics encompasses the study of vibrational waves in matter at any frequency. Objects capable of acoustic resonance typically possess multiple resonant frequencies, often corresponding to harmonics of the fundamental mode. When subjected to a complex excitation such as an impulsive force or broadband noise, these objects vibrate most strongly at their resonance frequencies while other frequencies are quickly attenuated, acting as a natural filter.

Principles of acoustic resonance and tuning fork interactions

A classic demonstration of acoustic resonance is the interaction between two tuning forks tuned to the same frequency. When one tuning fork is struck with a mallet, it generates periodic variations in air pressure and density. These pressure oscillations propagate through the surrounding air and can induce sympathetic vibration in the second tuning fork, even though it has not been directly struck. This sympathetic behaviour occurs because the incident sound wave matches the natural frequency of the unstruck fork. If a small piece of material is placed on one of the fork’s prongs, the added mass alters its natural frequency and introduces damping, resulting in weaker excitation and a diminished resonant effect.
Resonance can have both constructive and destructive implications. While desirable in musical acoustics for sound amplification and tone production, extreme resonant excitation may lead to structural failure. A well-known example is the shattering of glass when exposed to sound at precisely the glass’s resonant frequency, provided the amplitude is sufficiently large.
Acoustic resonance also plays a central role in biological hearing. Within the mammalian inner ear, the basilar membrane acts as a resonant structure whose mechanical properties vary along its length. High-frequency sounds induce maximum vibration near the stiff, narrow base of the membrane, whereas low-frequency sounds cause stronger movement near the wider, more compliant apex. This spatial separation of resonant frequencies enables frequency-specific activation of sensory hair cells, forming the basis of auditory perception.

Resonance in vibrating strings

Strings placed under tension, such as those found in guitars, harps, violins, pianos, and similar musical instruments, resonate according to their physical properties. The fundamental resonance of a string fixed at both ends corresponds to a standing wave with a wavelength twice the length of the string. Higher resonances, or harmonics, arise at wavelengths that are integer fractions of the fundamental. The associated resonant frequencies are expressed mathematically asf = nv / 2L,where f is the frequency, n is the harmonic number, v is the speed of the wave along the string, and L is the string length.
The speed v in a vibrating string depends on two additional properties: the tension T and the mass per unit length ρ, given byv = √(T / ρ).Substituting this into the earlier equation yieldsf = n √(T / ρ) / 2L,showing that higher tension and shorter length lead to higher resonant frequencies.
When a string is plucked or struck, the impulsive action excites many frequencies simultaneously. However, only those frequencies corresponding to the string’s resonant modes persist, as non-resonant components decay rapidly. String resonance also occurs sympathetically: if two strings share common overtone frequencies, one may vibrate in response to the other. A well-known example arises on string instruments where an A string at 440 Hz can cause an E string at 330 Hz to resonate, since both share a harmonic at 1,320 Hz.

Resonance in air columns and tubes

Air columns within tubes exhibit acoustic resonance strongly influenced by the tube’s length, bore shape, and boundary conditions at each end. Many wind instruments can be modelled as cylindrical or conical tubes that behave either as open pipes (open at both ends) or closed pipes (closed at one end). For example, orchestral flutes behave approximately as open cylindrical pipes, whereas clarinets act as closed cylindrical pipes, and double reeds such as oboes and bassoons function acoustically as closed conical pipes. Brass instruments similarly approximate closed conical systems, albeit with modifications arising from their flaring bells and mouthpieces.
In an air column, sound propagates as a longitudinal wave, generating alternating regions of compression and rarefaction. When reflected within the tube, these waves form standing wave patterns. At closed ends, particle motion is restricted, leading to displacement nodes and pressure antinodes. At open ends, the opposite occurs: particle displacement is large and a displacement antinode forms, accompanied by a pressure node. These boundary conditions govern the possible standing wave patterns and hence the resonant frequencies of the tube.
In tubes closed at both ends, the permitted wavelengths are given byλ = 2L / n,and the resonant frequencies byf = nv / 2L,where n is a positive integer and v is the speed of sound in air.
Open cylindrical tubes also follow this pattern, producing harmonics at frequenciesf = nv / 2L.However, a practical refinement must be made: the pressure node at the opening does not occur exactly at the end of the tube but slightly outside it. This is accounted for by an end correction of approximately 0.6 times the tube radius r, yielding a more accurate frequency expression:f = nv / 2(L + 0.6r).
Overblowing an open tube increases the excitation to allow a higher harmonic to dominate, often resulting in a pitch one octave above the fundamental. For instance, if the lowest note of an open pipe is C1, overblowing may produce C2, corresponding to the second harmonic.
Conical tubes differ subtly from cylindrical ones because their geometry allows true harmonic series even when closed at one end. This property underpins the tonal qualities of many orchestral woodwind and brass instruments.
Air columns also resonate in non-musical contexts. Cylinders closed at both ends, such as Rubens tubes, facilitate the visualisation of standing waves. In such devices, flames emerging through holes along the tube respond to pressure variations in the standing wave, creating a striking visual representation of the wave pattern.