06-07-2023 | Posted by Javier Rodríguez / David Guillén
Acoustics is the science of sound, including production, transmission, and biological and psychological effects. Sound sources may be airborne, caused by turbulent pressure fluctuations, or structure borne, caused by vibrations, the former typically involving higher frequencies. There are different methods to analyse transmission: wave acoustics, the more fundamental one, ray acoustics for larger distances, and energy acoustics for complex systems.
The equations governing wave transmission are conservation of mass and momentum; in addition, the equation of state relates pressure and density. This results in a speed of sound of about 340 m/s in normal conditions in air. Impedance is the relation between the pressure and the particle velocity, a non-uniform impedance causes reflections and refractions; diffraction is associated with changes in the wave front when going through holes or around obstacles.
Impedance design in musical instruments involves improving the match between the impedance of the vibrating system and that of the surrounding air. The first cavity resonators were large ceramic vases standing as decorations to absorb certain sounds. A related device is the Helmholtz resonator, a cavity with a broad exposed inlet and a small outlet.
Statistical Energy Analysis is useful for high frequencies in complex systems with numerous natural modes. It considers the mass and modal densities of individual systems, the internal loss factors (through a structural damping, inverse of the quality factor), and the coupling factors (relating power exchange to power difference).
Hearing reacts to intensity (energy flux). Since the Weber-Fechner law claims that organs respond in proportion to the logarithm of the stimulus, we define pressure level in decibels.
Sound may produce annoying reactions, for example:
In all cases it is generally better to try to control annoying sound and vibrations at the source.
Sounds may also create a pleasant experience, involving subjective and objective aspects. The (subjective) pitch is mostly related to the (objective) frequency, loudness to intensity, and timbre to waveform. The (liquid-filled) inner-ear cavity is excited by the action of dense small bones (hammer, anvil, and stirrup) ejecting waves with different paths in the cochlea depending on the frequency; the sensory receptors (hair cells) are in the basilar membrane.
Frequency is a key aspect in music. At very low frequencies we distinguish the successive beats and do not infer a pitch. The coherence of frequencies is what differentiates “sound” from “noise”. Also, when two notes of slightly different pitch are sounded simultaneously, we perceive beating. If the frequency difference between two signals is large enough, two resonance regions on the basilar membrane respond, and we hear the two original tones; our ability to analyse a complex pattern into its original pure tones is called frequency discrimination.
To achieve a harmonious effect, the Pythagorean tuning used intervals based on the ratio 3:2, the “pure” perfect fifth. In the equal-tempered scale, each octave is divided into twelve equal interval ratios; the notes are not as harmonious, but they are very close and allow playing in any key. Modern instruments are generally adapted to the tempered scale.
In Händel’s time (1685-1759), the reference note A4 was determined by his personal tuning fork, at a frequency of 422.5 Hz. Since brilliance increases with frequency, the standard A4 gradually went up. Eventually, in 1953 the International Standards Organisation recommended 440 Hz.
The character of a musical note may be enriched by producing periodic variations in pitch, loudness, or timbre at the rate of 4-8 Hz (vibrato for pitch and tremolo for loudness).
A certain reverberation in an auditorium is desirable and allows making concurrent comparisons of successive tones; an ideal reverberation time may be about 2 s.
