Resonance
In science, resonance describes the phenomenon of maximum energy transfer between oscillating systems, occurring as the vibration frequency of the energy-supplying system (the driver) approaches the natural frequency (or resonant frequency) of the receiving system (the resonator). Natural frequency: The frequency at which a system vibrates when it is initially supplied with energy and then left to vibrate freely, without further intervention. Think of pushing a swing, plucking a string, or striking a drum and then allowing each to vibrate on its own. When the driving frequency matches or approaches the resonator's natural frequency, energy transfer is maximized, allowing vibration amplitude to build up. Experiment with this interactive applet (requires Java plug-in). Resonance can produce dramatic effects, including the collapse of structures (e.g., the Tacoma Narrows Bridge) or the shattering of a glass by a singer's voice (e.g., here and here). [ More details on the Tacoma Narrows Bridge collapse, assessing resonance as its initiator rather than the sole cause: here and here ] In musical instruments: Sound generators (e.g., strings, drumheads, reeds, metal bars) vibrate at their resonant/natural frequencies when excited. Resonators (e.g., body and air cavity of a guitar or violin; the air columns of a clarinets, saxophones, trumpets, and organ pipes; the tubes beneath vibraphone bars) enhance vibrations at frequencies matching or approaching their own natural frequencies, effectively amplifying selected components of the sound.
Natural (resonant) Frequencies of Vibrating Bodies The natural frequencies of physical objects depend on their characteristics, including size, shape, mass, density, and stiffness/tension. General guidelines: Length: The longer a string (e.g., on a guitar) or air column (e.g., inside a trombone), the lower the natural frequency. In general, frequency is inversely proportional to length. For vibrating wooden or metal bars, frequency is approximately inversely proportional to the square of length.   Mass: The greater the mass of a string or drumhead, or the larger the air volume within certain resonators, the lower the natural frequency. In general, frequency is inversely proportional to the square root of mass, all other factors being equal.   Tension or stiffness: The greater the tension of a string or drumhead, or the greater the stiffness of a vibrating material (e.g., xylophone or vibraphone bars), the higher the natural frequency.  In general, frequency is proportional to the square root of tension or stiffness, all other factors being equal.   Shape and symmetry: More symmetrical shapes with fewer irregularities often produce simpler vibration patterns and are more likely to support more harmonic spectra. However, harmonicity also depends on the material and boundary conditions. For example: a) An ideal, cylindrical air column excited by an air stream [ e.g., flute, pan flute, organ pipe (may deviate from ideal cylinder) ] produces a largely harmonic spectrum, often dominated by its fundamental frequency. b) A struck cymbal produces a predominantly inharmonic spectrum, with prominent higher-frequency components. Most musical instruments incorporate resonators whose shapes, materials, and construction support multiple natural frequencies. This allows them to respond to several frequency components of a single note and to multiple notes. Musical resonators are therefore generally broadly tuned: rather than amplifying a single frequency and its integer multiples, they enhance vibrations across a relatively wide range of frequencies, ideally covering the instrument's frequency range. However, this reduces the maximum amplification possible; there is an inverse relationship between frequency response range and level.
Types of Resonance
Conductive (Forced) Resonance  A resonator (e.g., guitar body) begins vibrating because it is physically connected to a vibrating source (e.g., strings, through the bridge). The source transfers energy directly to the resonator, forcing it to vibrate. [ Illustrations of exaggerated vibration modes of a guitar body in response to string vibrations:   Animation 1  -  Animation 2  -  Simulation ]	Sympathetic (Free) Resonance A resonator (e.g., xylophone resonant tube) begins vibrating in response to wave energy reaching it through the air from another vibrating object (e.g., struck wooden bars). [ Animation    -    Video ]
Weak Coupling and Synchronization Between the extremes of:    a) Direct, strong physical coupling (conductive resonance) and    b) No direct physical contact (air-mediated sympathetic resonance), there is weak coupling, in which oscillators exchange energy through a relatively weak physical connection (e.g., pendulums connected by springs). Weak coupling supports slow, low-level energy transfer and can lead to synchronization among oscillators [ Video: Metronome synchronization   -   optional details ]. There are multiple musical instrument examples of weak coupling between generator and resonator, to be discussed later in the module.   Resonator Response and Damping The figures below illustrate how the vibration amplitude (vertical, y axis) of a resonator with natural frequency f0 changes with driving frequency (horizontal, x axis) and with different degrees of damping (i.e., friction, resistance, or energy loss). [ Figure below-right:    f0 =440Hz;   blue: low damping;  green: medium damping;  red: high damping ].
Video: Relationship between driver frequency and resonator response. [ Additional short video ]	Key Observations on Resonator Response   _ Maximum response occurs when the driving frequency is at or near the resonator's natural frequency.   _ Response is generally greater for driving frequencies below than above the natural frequency (when comparing driving frequencies equidistant from natural frequency). At sufficiently low driving frequencies, the resonator approximately follows the driver's motion.   At driving frequencies well above resonance, the resonator's response decreases rapidly, approaching zero. _ Increasing damping generally reduces the maximum response, broadens the resonance peak, and lowers the frequency at which maximum displacement occurs.   In general, the broader the frequency response, the lower the maximum response amplitude (why?)
Resonance and Filters
Resonators behave similarly to filters. Both selectively amplify or attenuate different frequencies according to their physical, electronic, or digital characteristics. The figures below illustrate the four most common types of idealized filters.
Resonator/Filter Q factor, Bandwidth, and Damping; The Q factor (Quality factor) describes the sharpness of a resonator's or filter's frequency response.                         Q = = Resonant Frequency /  Response Bandwidth  where bandwidth is the difference between the upper and lower frequencies outlining a specific response range. _Greater damping produces a broader response curve and a lower Q factor. _Lower damping produces a narrower, sharper response curve and a higher Q factor.
Measuring Filter Bandwidth (figure to the right) 3dB Response Bandwidth: The frequency range between the two points on the curve where the response has dropped by 3dB relative to its reference or peak level. Note: for electronic filters, a 3dB decrease corresponds approximately to: _ A 50% reduction in intensity (or power) but _ A 29% reduction in voltage amplitude [ optional explanation ] 10dB Response Bandwidth: The frequency range between the two points on the curve where the response has dropped by 10dB relative to its reference or peak level. Filter slope: The rate at which a filter's response level changes with frequency, commonly expressed in decibels per octave (dB/octave). The most most effective slope that is easy to implement produces a 12 dB/octave roll-off. It is the simplest slope that cuts off unwanted frequencies fast enough without requiring overly complex or expensive multi-stage builds. [ Filer slopes that produce 12 dB/octave roll-off are achieved by a 2nd-order network, typically combining two reactive components like inductors and capacitors. ] Why do you think 3dB and 10dB are commonly used as reference points when describing filter response?

 

Microphone Frequency Response The figures below, called frequency response curves, show how the output level of two popular vocal microphones varies with frequency. They resemble spectral envelopes: continuous curves indicating the relative response at different frequencies, rather than displaying response at individual frequencies. Frequency response curves reveal: How a microphone responds mechanically (through its diaphragm) and electronically (through its circuit) to incoming sound waves. How its sensitivity varies across frequencies. An ideal microphone would have a perfectly flat frequency response extending across the entire human hearing range (approximately 20-20,000Hz), with no frequency-dependent boosts (resonances) or reductions (damping). Its response curve would therefore be a horizontal line at or near 0dB (relative to a reference sensitivity). Such a response is difficult to achieve in practice (why?). Microphones designed for accurate (transparent) sound reproduction aim to approximate this ideal. A microphone with a nearly flat, wide-range frequency response: Captures sound more faithfully, without artificially emphasizing or suppressing particular frequencies. A smoother response can also reduce susceptibility to feedback at pronounced resonant peaks (why?). Accommodates a broad range of sound sources, including those containing substantial energy at very low or very high frequencies. [ Animations: How microphones work -  outline  -  dynamic microphones  -  condenser microphones ]

Figure: Frequency response curves: Oktava 319 (top) and Sure SM58 (bottom)

 

Standing Waves (Resonance) of Stings, Air Columns, and Plates
Standing waves are caused by resonance in bounded or reflective systems and develop at the natural frequencies of vibrating systems.   Standing Waves on Strings The harmonic spectral components, or harmonics, of tones produced by standing waves on a string correspond to wavelengths λ directly related to the string's length L. For a string (fixed at both ends), each loop represents half a wavelength. Therefore, the first three modes of vibration (and the corresponding harmonics) have wavelengths: λ1=2/1L;    λ2=2/2L;    λ3=2/3L. Equivalently:  L = 1/2λ1 = 2/2λ2 = 3/2λ3. The first mode is the fundamental, while the second and third modes correspond to the second and third harmonics.
Above: Relationship among string length, wavelength, and mode of vibration (harmonic number) for ideal strings. To the right: Two resources illustrating the formation of sawtooth standing-wave patterns on strings, initiated through conductive (forced) excitation.
Standing Waves in Air Columns Similarly, standing waves inside air columns produce resonant frequencies, whose wavelengths depend on the column length L, shape (e.g., cylindrical or conical), and boundary conditions (e.g., open or closed ends), and form loops that represent half a wavelength. For idealized air columns, the permitted vibration patterns differ according to whether the column is open or closed at its ends. Cylinder Open at Both Ends [ below left - flutes, pan flutes, some organ pipes, etc. ]:    The first four harmonic modes of vibration have wavelengths:  λ1=2L;  λ2=1L;  λ3=2/3L;  λ4=1/2L. All integer harmonics are supported. Cylinder Closed at One End [ below middle - clarinet (approximately), several non-Western woodwinds, etc. ]:    The first four modes have wavelengths: λ1=4L;   λ2=--;   λ3=4/3L;   λ4=--. Only odd harmonics are supported in the idealized model. Cone Closed at One End [ below right - oboe, saxophone (approximately), etc. ]:    The first four modes have wavelengths: λ1=2L;   λ2=1L;   λ3=2/3L;   λ4=1/2L. All integer harmonics are supported.
[ Figure: Standing-wave patterns in open cylinders, closed cylinders, and conical air columns. ]   NOTE: In real instruments, these relationships are more complicated because: _ A vibrating string undergoes small changes in length and tension during excitation. _ The effective length of an air column (i.e., the air column length that needs to be used in any calculations, which is different from its physical length) depends on its diameter, shape, and openings. Nonetheless, for most practical purposes, assuming equal lengths and comparable wave speeds: Stings, conical air columns (e.g., saxophone), and cylindrical air columns open at both ends (e.g., flute) have similar fundamental wavelengths and frequencies. Cylindrical air columns closed at one end (e.g., clarinet) have a fundamental wavelength approximately twice as long, producing a frequency one octave lower than an open cylinder (e.g., flute).
Standing Waves on Metal Plates and Membranes
Metal plates, such as cymbals (below and bottom-right), and circular membranes, such as drumheads (top-right), vibrate in complex patterns that generally produce signals with inharmonic spectra: their component frequencies are not integer multiples of the lowest resonant frequency. Consequently, they typically produce sounds without a definite musical pitch. Instruments like the Timpani (hemispherical drum, also called kettledrum) achieve a distinct pitch because the coupled enclosed air volume and the shell shift a specific subset of lower modes --which last longer than higher, inharmonic modes-- closer to an integer multiple relationship. Similarly, drums that include a bottom skin can be tuned to achieve a comparable effect. Instruments like the Crotales (small, thick, metal discs, also called antique cymbals) achieve a distinct pitch because their high thickness-to-diameter ratio, flat middle area, and raised center suppress higher-frequency components and reinforce a clear, single, low-frequency mode. Although the relationship between a cymbal's or membrane's dimensions and its resonant wavelengths is complex, its vibrations still follow specific, identifiable patterns or modes of vibration (click on the images for more).	------------------------------------ [ Video by the Fluke Corporation ]

 

Helmholtz Resonators and Bass Reflex Loudspeakers
Helmholtz resonators, named after 19th century physicist and physician Hermann von Helmholtz, are containers with an enclosed air volume connected to the outside through a relatively short, narrow opening or neck.  Their operation resembles a mass-spring system:      _ The air inside the container behaves like a spring.      _ The air in the neck behaves like a moving mass.      _ The system's approximate resonant frequency is expressed by the formula to the right.	where          f: resonant frequency          c: speed of sound          A: cross-sectional area of the neck          l: neck length          V: container volume
How Helmholtz Resonance Works
Consider a small lump of air occupying the neck of the bottle (shaded in the diagrams). _ An air jet pushes this lump slightly into the bottle, compressing the air inside. _ The increased internal pressure pushes the lump back outward. Its momentum carries it beyond its original position. _ This outward motion lowers the pressure inside the bottle (rarefaction), drawing the lump back inward. _ The process repeats, producing an oscillation.  When blowing across a bottle opening, the air jet from your lips can alternately deflect into and away from the opening, supplying the energy needed to sustain the vibration. The resonant frequency depends on the speed of sound c, the dimensions of the neck (length l and area A), and the container's volume V, per the above formula. For this mass-spring-like system:      _ Longer necks ( l ) produce lower resonant frequencies.      _ Larger container volumes (V) produce lower resonant frequencies      _ Wider necks (larger A) produce higher resonant frequencies.   Applications: Bass-Reflex Loudspeakers Helmholtz resonance is used in applications ranging from narrow-band sound absorption (see further below) and to low-frequency enhancement in loudspeakers. A bass-reflex loudspeaker incorporates a tuned opening (or port) that improves low-frequency efficiency and extends the system's useful bass response. It consists of: An enclosed air cavity of carefully selected volume. An opening or tube connecting the inside of the loudspeaker enclosure to the outside, with dimensions chosen to support resonance within a desired low-frequency range. The enclosed air behaves like the air inside a Helmholtz resonator, while the air in the port behaves like the lump of air inside the resonator's neck. The enclosure and port are tuned together so that their resonance enhances low frequency output, allowing the loudspeaker to reproduce frequencies below the range it would otherwise reproduce efficiently. In addition, the port provides an additional path for acoustic energy to radiate outward. More details on bass-reflex loudspeaker design here and here. [ Animations: How loudspeakers work --- overview  -  component breakdown  -  cone and surround ]

SOUND PROPAGATION Reflection - Transmission - Diffusion/Reverberation - Absorption
When a sound wave encounters a boundary, either within a medium or between two different media (1 and 2), its energy may be: _Reflected: Bounced back toward the original medium, potentially producing standing waves, echoes, or reverberation. _Transmitted: Passed through the boundary. Depending on the boundary and surrounding conditions, transmitted sound energy may also be:    _ Absorbed: Converted into other forms of energy (primarily heats).    _ Diffracted: Bent around obstacles or spread through openings.    _ Refracted: Bent as it enters a medium where sound travels at a different speed.
Reflection Acoustics borrows the basic law of reflection from optics: The angle of reflection equals the angle of incidence, both measured relative to the reflecting surface's normal*. [ *Normal: A line perpendicular to the surface at the point of incidence. In three dimensions, the incident wave, reflective wave, and surface normal lie in the same plane. ]
The proportion of sound energy reflected versus transmitted at a boundary depends primarily on the acoustic impedance mismatch between the two media. In general, the greater the impedance mismatch and the larger the angle of incidence, the greater the reflected energy. Practically all musical instruments rely on reflection to establish standing waves and produce sound. They achieve this through structures with appropriately positioned boundaries and impedance mismatches. If: z1: Acoustic impedance of the medium from which the sound wave originates. z2: Acoustic impedance of the medium toward which the sound wave travels. Then (for pressure waves at normal incidence): If z1 = z2 (not displayed): No reflection; complete transmission. If z2 < z1 (below, top-right): The reflected pressure wave is 1800 out of phase with the incident wave.                                               [ however, the reflected displacement wave is in phase with the incident wave ] If z2 > z1 (below, bottom-right): The reflected pressure wave is in phase with the incident wave.                                               [ however, the reflected displacement wave is 1800 out of phase with the incident wave ] If z2 <<< z1 (below, top-left) or z2 >>> z1 (below, bottom-left):  Nearly total reflection; negligible transmission. Note: _ Phase reversal at free versus fixed ends depends on whether pressure or particle displacement is being examined.           _ Reflected plus transmitted energy = incident energy. OPTIONAL: Quantifying reflected and transmitted energy. If     I0: Incident Intensity;   Ir: Reflected intensity;  and  It: Transmitted intensity;    then, at normal incidence:     Proportion of Reflected Intensity   = Ir / I0 =   [(z2-z1) / (z2+z1)]2       Proportion of Transmitted Intensity = It / I0 = 1 - Ir / I0 = 1 -  [(z2-z1) / (z2+z1)]2  = 4z2z1 / (z2+z1)2  [ Detailed derivation. Remember that intensity is proportional to the square of sound pressure for a given medium. ]

Examples of reflection - dependence on impedance [ From the University of Saskatchewan, Department of Engineering Physics ]
Z2 = 0   --   Total reflection at a free end. The reflected and incident displacement waves are in phase, while the reflected pressure wave reverses phase.	Z1 / Z2 = 2   --   The second medium has lower impedance. The reflected and incident displacement waves are in phase, while the reflected pressure wave reverses phase.
Z2 = infinity   --   Total reflection at a fixed end. The reflected displacement wave reverses phase, while the reflected and incident pressure waves are in phase.	Z1 / Z2 = 0.5   --   The second medium has higher impedance. The reflected displacement wave reverses phase, while the reflected and incident pressure waves are in phase.

Diffusion / Reverberation Successive, partial reflections from multiple surfaces in an enclosure --such as the uneven, complex-shaped walls of an auditorium, scatter sound waves in different directions, producing diffusion of sound energy (see below). When these reflections become sufficiently numerous and diffuse, their combined, gradually decaying sound is perceived as reverberation.    Reverberation is a desirable property of auditoriums to the extent that it:     a) helps compensate for the inverse-square-law decrease in direct sound intensity with distance;     b) reduces overall and frequency-specific sound-level differences across the enclosure; and     c) contributes to the distinct sonic character associated with different musical styles.
Reverberation time (RT60) is the time required for the reverberant sound level to decrease by 60dB after the sound source stops. Explore qualitatively the main concepts presented in these two resources: Reverberation & Reverberation Time (HyperPhysics, Department of Physics and Astronomy, Georgia State University). You are not expected to follow their additional links or perform any calculations. More on reverberation later in the course.
Absorption Absorption describes the loss of sound-wave energy as sound waves propagate through a medium or interact with a boundary. Sound energy may be absorbed through: a) Friction: As sound propagates through air or other materials. b) Porous or flexible materials: As sound interacts with an moves through their structure. c) Thermal losses: As acoustic energy is converted into heat.  d) Layered structures: Through mechanical damping, interaction, and decoupling between layers.
	The effectiveness of a material or structure in absorbing sound is expressed by its absorption coefficient (a). Absorption coefficients:    _ Generally range from 0 (no absorption) to 1 (complete absorption).    _ Depend on frequency.    _ Depend on the material's composition and physical structure. [ Brief qualitative overview of absorption and table of absorption coefficients for common materials. ] [ Additional table of absorption coefficients. ] Types of sound absorbers Velocity absorbers (porous materials such as foam or fiberglass) absorb sound primarily through friction associated with air-particle movement. They are broadly tuned and generally most effective at mid and high frequencies, which develop increasingly higher particle velocities. Pressure absorbers (tuned membranes and resonant traps, such as Helmholtz resonators) absorb sound through resonant motion energy dissipation, interference, and diffusion. They are typically more narrowly tuned and particularly effective at low frequencies, especially near reflective boundaries where sound-pressure variations are high. [ Optional: additional details on tuned membrane absorbers ] Hybrid absorbers (such as modified Helmholtz absorbers and multilayered drapes of varying mass, density, and spacing) combine characteristics of velocity and pressure absorbers to achieve substantial absorption over a broader frequency range. However, such systems may be expensive and and require considerable installation space or specially developed materials. [ Optional: additional details on (i) layered Helmholtz absorbers here & here and (ii) multilayered drape absorbers here & here ]

Diffraction - Refraction - The Doppler Effect [ special cases of sound-wave propagation ]
Diffraction Diffraction describes the ability of sound waves to spread and bend around obstacles and through openings whose dimensions are comparable to or smaller than the sound waves' wavelength. Diffraction plays a major role in determining: Radiation patterns: How sound spreads outwards from sources such as musical instruments and loudspeakers. Pick-up patterns: How sound is received by sensors, such as microphones, and receptors, such as the human ear. Sound propagation through spaces: How sound travels around obstacles and through openings.  General principle Low frequencies diffract more readily than high frequencies. This is because low-frequency sounds have longer wavelengths, making everyday obstacles and openings relatively smaller compared with the sounds' wavelengths. Diffraction becomes: _ appreciable when the wavelength exceeds roughly half the size of the opening or obstacle (or, for obstacles/openings <~ twice the wavelength). _ very pronounced when the wavelength is approximately four times the obstacle or opening size, or greater (or, for obstacles/openings <~ 1/4 wavelength). For example, low-frequency wavelengths are typically much larger than the subwoofers producing them. Consequently, individual subwoofers tend to radiate low-frequency sound approximately equally in all directions, functioning as point sources. [ Could we somehow control this radiation pattern and make subwoofers directional? ] Diffraction around obstacles versus through openings Similarity: _ The smaller the obstacle or opening, relative to wavelength, the wider the spread of sound-wave energy around/through it. Difference: _ The smaller the obstacle the larger the proportion of total sound-wave energy that will be transmitted around it. _ The smaller the opening, the smaller the proportion of total sound-wave energy that will be transmitted through it.    Explore this relationship using the diffraction simulation. Select the slits option and experiment with the frequency and slit-width faders. Diffraction examples at different wavelengths [ Assume c = 345m/s @ ~210C ] The dimensions in the examples below are approximate illustrations of wavelength-dependent behavior, not sharp physical thresholds. In practice, diffraction and sound blockage also depend on obstacle shape, geometry, and surrounding conditions. Example A: f = 66Hz Approximately the fundamental frequency of a C2 note played on a bass guitar.      Period T = 1/66 s/cycle;          ---       λ = c*T = c/f = 345/66 => λ=5.23m/cycle At this frequency:   _ Diffraction becomes appreciable for openings/obstacles <≈10.46m, and very pronounced <≈1.31m.   _ Sound blockage by obstacles becomes increasingly significant  >≈10.46m and becomes substantial >≈20.92m   Example B: f = 528Hz Approximately the fundamental frequency of a C5 note played on a flute (or the frequency of the 8th harmonic of the bass guitar note).     Period T = 1/528 s/cycle;        ---        λ = c*T = c/f = 345/528 => λ=0.65m/cycle At this frequency:   _ Diffraction becomes appreciable for openings/obstacles <≈1.30m, and very pronounced <≈0.16m.   _ Sound blockage by obstacles becomes increasingly significant  >≈1.30m and becomes substantial >≈2.60m Diffraction and sound quality Most sounds correspond to complex signals and have spectra with multiple frequency components. Consequently, a given obstacle or opening may diffract some frequencies more strongly than others. The sound spectrum on the other side of the obstacle/opening --and therefore its perceived sound quality-- depends on the position of a listener relative to the sound source and the obstacle/opening. More on this during the module on timbre.
Illustrations of diffraction and its dependence on frequency [ from HyperPhysics, GSU ]
Refraction Refraction describes a change in the propagation direction of sound waves as they pass between media --or regions of a medium-- in which sound travels at different speeds [ see Refraction, HyperPhysics, GSU ] Optional: Snell's law of refraction: sinθ1 / sinθ2 = v1 / v2 The laws of reflection and refraction can be derived from Fermat's principle of least time. Refraction and diffraction can be explained through the Huygens-Fresnel wavefront principle and Bernoulli's principle of linear superposition. In the figure to the right: v2 < v1 => the sound waves refract/bend towards the normal (the dotted line perpendicular to the boundary). If  v2 > v1, sound waves would bend in the opposite direction, away from the normal. Real-world example: Outdoor sound propagation The speed of sound in air depends on temperature and on wind speed and direction. Consequently, sound waves may be refracted as they travel through air layers with different temperatures (e.g. as they rise above the ground), wind speeds, or wind directions. For example, temperature and wind conditions above the ground can bend sound waves upward or downward, significantly affecting how far they travel and where they are heard. This makes the orientation of outdoor performance venues relative to prevailing winds and expected temperature gradients an important consideration for: _ Efficient sound transmission toward the audience and _ Minimizing unwanted sound propagation into surrounding communities.	Explore refraction using this simulation.
Figure: Illustrations of the refraction effects of temperature and wind differentials.
The Doppler effect The Doppler effect, named after 19th century Austrian physicist Christian A. Doppler, describes the change in observed frequency caused by relative motion between a wave source and a receiver. It has important applications in acoustics, optics, astronomy, medicine, audio technology, and many other fields. For example, when a sound source moves relative to a listener, the emitted frequency will appear higher (as the source approaches) or lower (as the source recedes) than the emitted frequency, because wave fronts are compressed in-front of a moving source and are spread apart behind it. _ HyperPhysics resource on the Doppler effect - approach any equations qualitatively [ alternative, multimedia-rich explanation ] A popular musical application of the Doppler effect is in the Leslie Organ Loudspeaker, producing a distinct frequency modulation (vibrato) effect. If the source moves faster than sound, wave-fronts will pile up and add behind the source, resulting in the "sonic boom" effect experienced, for example, when a supersonic airplane breaks through this wall of compressed air. Notice how you can barely hear the plane approaching [ more detailed video  -  optional details ].
Illustrations of the Doppler effect - dependence on source speed (from the University of Saskatchewan, Department of Engineering Physics)
Source velocity = 1/2 * Sound velocity	Source velocity = 2 * Sound velocity

Musical Instruments as Sound Generators and Transmitters
HISTORICAL APPROACH
The most common (but highly inconsistent) everyday classification of musical instruments divides them into three categories: _ Percussion _ Winds (brass and woodwinds) _ Strings
A more systematic classification originated with Victor-Charles Mahillon, musician, instrument builder, and curator of musical instruments at the Brussels Music Conservatory, in the late 19th century. Mahillon's system was later modified and expanded by Erich von Hornbostel and Curt Sachs [ optional: the Hornbostel-Sacks system ] to accommodate musical instruments from cultures beyond the Western musical tradition.
The system includes five principal categories  --primarily (but not always) based on what initially vibrates to produce sound-- and multiple levels of subcategories, totaling over 300 instrument groupings:		Chordophones: Sounds is produced primarily by vibrations of one or more stretched strings, involving one-dimentional standing waves (guitars, harps, pianos, violins, etc.).
Aerophones: Sound is produced primarily by vibrations of air, typically within a volume defined by the instrument, involving standing waves in air columns or cavities (saxophones, oboes, trumpets, organs, etc.).		Membranophones: Sound is produced primarily by vibrations of a stretched membrane, involving two-dimensional standing waves (drums, kazoos, etc.).
Idiophones: Sound is produced primarily by vibrations of the instrument's body, involving three-dimentional vibration patterns on the entire instrument (cymbals, claves, cowbells, etc.).		Electrophones: Sound is produced primarily through electrical oscillations of electronic circuit (synthesizers, theremins (explanation), etc.).
ACOUSTICS APPROACH
From and acoustics perspective, musical instruments are examined systematically in terms of: Their physical properties How energy is supplied, transferred, modified, and radiated. The resulting auditory perceptions. [ See Hajda, J.M. (2009). Psychoacoustics and Organology. Journal of the Acoustical Society of America, 126: 2236 ] This course examines musical instruments  as  driver - generator - resonator  systems, incorporating different forms of: Coupling: Facilitating energy transfer    a) from outside to inside the instrument (e.g., mouthpieces);    b) between components within the instrument (e.g., bridges); and    c) from the instrument to the surrounding air (e.g., vibrating surfaces, openings, or bells). and with different types of Articulation: Controlling how vibrations are    a) initiated (e.g., striking, blowing, plucking);    b) modified in pitch (e.g., stopping a string at different positions);    c) maintained (e.g., bowing, blowing); and    d) terminated (e.g., with or without performer intervention).  Voice acoustics uses slightly different terminology:: Actuator (driver) - Vibrator (generator) - Resonator - Articulator(s) Beyond the terminology differences in the first two stages, the principal distinction is that, in voice production, articulation occurs primarily at the end of the sound-production process, also facilitating the coupling of sound into the surrounding air. In most other instruments, articulation can occur at various stages.
The Generator The heart of any musical instrument is its sound generator. The generator is the component whose vibrations initiate the sound -production process when excited by a driving force. These vibrations produce sound waves that propagate through the instrument and into the surrounding air, eventually reaching our ears or a microphone. Examples include: Strings (violin, piano,...);     Reeds (clarinet, saxophone,...);     Air (flutes, organ flue pipes,...);     Lips (trumpet, trombone, bugle,...); Membranes (various drums,...);     Metal plates/bars (cymbals, gongs, vibraphones, );     Vocal folds (human voice).  The generator's vibration characteristics help determine frequency and spectral ranges of the sounds produced. Variable  versus Fixed Generators Some generators are variable. For example, strings can be stopped"(pressed against a fret or fingerboard), changing their effective vibrating length. Similarly, modifying the pressure and position of a player's lips against a saxophone reeds can alter its vibration characteristics. Such changes affect the generator's physical and resonant properties, modifying the pitch and/or timbre produced. Other generators are fixed. Examples include instruments with fixed-length strings (pianos, harps) and various percussion instruments.  Exceptions: mechanisms that allow pitch modification [ e.g., piano string tuning; harp pedals/levers ] or variable membrane tension [ e.g., tabla or certain toms ].
The Driver The driver excites the generator by applying (articulating) a force that initiates or sustains (drives) its vibrations. This force can vary in duration, modulation, magnitude, and location, influencing the resulting sound. Duration of the Driving Force Instantaneous or short-duration excitation: Plucking, striking, etc.    For short-duration excitation, the force may be applied over different contact times. Examples:       _ Plucking a string with the fingertips (longer contact) versus a plectrum (shorter contact).       _ Striking a drum with a felt-tipped (longer contact) versus a wooden-tipped (shorter contact) mallet .      In general, the shorter the contact time and the more rapid the energy transfer:       a) the more abrupt, inharmonic, and perceptually significant the attack portion of the signal envelope.       b) the richer the resulting spectrum in high-frequency components. Modulation of the Driving Force Continuous excitation: Bowing, blowing, etc.    During continuous excitation, the driving force may be modulated in various ways, producing effects such as:       _ Vibrato: Small, usually periodic variations in frequency/pitch (e.g., string bending or fast trills).       _ Tremolo: Usually periodic, narrow variations in amplitude/loudness modulation, or rapid repetitions of a note.       _ Portamento: Continuous sliding between pitches.       _ Glissando: Sliding through a succession of pitches in discrete steps (portamento and glissando are sometimes used interchangeably).   The more a signals varies over time, the more likely it is to attract attention and stand out among simultaneous sounds (i.e., to cut through a complex sonic texture). Location of the Driving Forcer In the middle (i.e., away from any support) vs. towards the edges (i.e., near a support).    _ Exciting a generator (such as a string, membrane, or bar) closer to its middle favors lower frequency vibration modes,       producing a spectrum relatively richer in low-frequency components.    _ Conversely, exciting a generator closer to its fixed edges does not enhance low frequency modes,       shifting the spectral balance towards higher frequencies. In General The magnitude of the driving force is closely linked to the dynamic range of the sounds produced The manner in which the driving force is applied is closely linked to the signal and spectral envelopes of the sounds produced.
The Resonator A generator is coupled to (connected to / placed near) a resonating system that can amplify, sustain, and spectrally shape its vibrations. Resonators contribute to a musical instrument's frequency response*.    [ *Frequency Response: the extent to which an instrument's resonating system enhances or attenuates different frequencies within a range of interest. ] Examples include:    _ Soundboards: Pianos and most acoustic string instruments.    _ Air cavities: Lutes, violins, drums; thumb pianos (e.g., mbira/kalimba), etc.    _ Air columns: Horns, pipes, tubes, etc.    _ Vocal tract: Human voice.    _ Mouth/sinuses: Voice, jew’s harp; harmonica, etc.    _ Membranes/plates: Resonant drumheads, plates, etc.  The frequency response characteristics of a resonator influence the produced sound and depend on resonator size, shape, material, and construction.  The resonating systems of musical instruments are preferably broadly (rather than sharply) tuned. That is, they preferably have an evenly-distributed frequency response, over a broad range of frequencies. In aerophones with variable resonators, air columns can be stopped (e.g. by pushing a trumpet valve or by gliding the outer slide tube of a trombone) to broaden the instrument's frequency response and pitch range.   Musical instrument resonators are therefore designed so that they can more-or-less evenly respond at and amplify a wide range of frequencies, usually matching the fundamental frequency range of the instrument to which they are attached, as well as key upper harmonics.
Couplers (e.g. various types of bridges or, in the case of coupling through proximity, air) link generators and resonators, essentially modifying their degree of impedance matching/mismatching to control the balance between transmitted energy onto the resonator and reflected energy back to the generator . The higher the degree of impedance matching between generator(s) and resonator(s) the higher the amount of energy transfer between them and the more efficient the coupler(s). There is always some degree of energy feedback between generators and resonators that depends on coupling type and the note produced. Consequently, generators and resonators may alternate roles and sometimes do so in a periodic/regular manner. The degree of such feedback and the nature of its periodicity determine, to an extend, the balance between an instrument’s radiation efficiency (see below) and decay time (i.e. how long it takes for the vibrations of the generator/resonator to die out after energy supply has stopped).  The difference, for example, between the decay times of a hammer dulcimer string and a classical guitar or a piano string reflects, in part, the different degree and periodicity of feedback between their respective generators and resonators, which depends on coupling (i.e. bridges). Note: For several idiophones (e.g. cymbals), generators and resonators coincide (e.g. metal plate). In such cases, articulation fulfills the function of coupling "in" (i.e. linking between generator and resonator). An instrument’s coupling to the atmosphere (i.e. the way sound waves are transferred from the instrument to the surrounding air; e.g. via the bell on a horn) influences the spectral characteristics of the sound leaving the instrument, as well as its radiation pattern. Radiation pattern: Term describing the way energy radiated from a sound source spreads away from it. Generally speaking, high frequencies are more directional (they follow a narrower path) than low frequencies, which spread more or less spherically (remember the discussions on the inverse-square law and on diffraction). Radiation efficiency: Term describing the ratio of acoustic pressure just outside the end of an instrument to that just inside.  Radiation efficiency varies according to instrument construction and the note being produced and is ultimately a function of the impedance difference between the inside and the outside of an instrument, which is modified by the coupler "out." In general, high frequencies radiate more efficiently than low frequencies. The radiation efficiency and radiation pattern of a musical instrument dictate the relationship between    a) the position of a listener or microphone relative to the instrument and    b) its perceived timbre.
Musical Instrument Resources
Standard online resources addressing multiple instruments include:    • Music Instrument Guides; Yamaha    • Instrumentology; Vienna Symphonic Library    • Musical Instruments; HyperPhysics    • Music Acoustics; University of South Wales    • The Physics of Music and Musical Instruments by David R. Lapp; Tufts University   Voice    • http://hyperphysics.phy-astr.gsu.edu/hbase/Music/voice.html  (outline)    • http://newt.phys.unsw.edu.au/jw/voice.html  ("Introduction and overview" section up to the "Source-filter interactions" subsection)    • https://voicefoundation.org/health-science/voice-disorders  (resource on voice production, health, & conservation)    • The Acoustics of the Singing Voice  (seminal 1977 paper by pioneer in voice acoustics, J. Sundberg)    • Research on the Singing Voice in Retrospect  (follow-up by J. Sundberg to his 1977 paper)    • https://www.youtube.com/watch?v=ZLgAQTMgZ6g  (voice production acoustics - video)    • https://muse.union.edu/griffiths-capstone  (Modeling the Human Vocal Tract - 2014 student project by S. Griffiths)    • https://youtu.be/UkNDBQmEMSY  (advanced review of formants; includes a link to a voice synth, based on formant manipulation - video)   Guitar    • http://hyperphysics.phy-astr.gsu.edu/hbase/Music/guita.html  (outline)    • https://newt.phys.unsw.edu.au/music/guitar/  (focus on "Basics about sound ...", "Anatomy of an acoustic....", & "Acoustics and components")    • https://youtu.be/gbP6h0dtcQw  (guitar acoustics - video)    • https://youtu.be/lBAZepM5F_0  (electric guitar fundamentals - video)   Marimba    • http://hyperphysics.phy-astr.gsu.edu/hbase/Music/xylo.html  (outline)    • https://en.wikibooks.org/wiki/Acoustics/Basic_Acoustics_of_the_Marimba  (short analysis)    • https://www.vsl.info/en/academy/percussion/marimba  (construction; sound production; etc.)    • https://www.youtube.com/watch?v=pCnFE-lSta4  &  https://www.youtube.com/watch?v=Bsj-xx25dHYY  (building a marimba - videos)    • https://www.youtube.com/watch?v=RaYvMwQd3cs  (composition for six marimbas by Steve Reich - video)    • https://www.jstage.jst.go.jp/article/ast/22/3/22_3_177/_pdf  (acoustics of percussion instruments - T.D. Rosing, 2001)   Trombonee    • http://hyperphysics.phy-astr.gsu.edu/hbase/Music/trombone.html  (outline - details available)    • http://newt.phys.unsw.edu.au/jw/brassacoustics.html  (info on all brass instruments, including the trombone)    • https://acousticstoday.org/wp-content/uploads/2016/12/Brass.pdf  (detailed outline of brass instrument acoustics)    • https://www.youtube.com/watch?v=CoxnhjLMVBo  (lip vibrations in trombone embouchures in 3 parts - video)   Flute    • http://hyperphysics.phy-astr.gsu.edu/hbase/Music/flute.html  (outline)    • http://www.markshep.com/flute/Acoustics.html  (outline)    • http://newt.phys.unsw.edu.au/music/flute/  (detailed discussion and resource database)    • http://www.flutepage.de/deutsch/goodies/physik.php?englisch=true  (detailed discussion)    • https://ccrma.stanford.edu/~pdelac/research/MyPublishedPapers/Thesis.pdf  (doctoral dissertation; P.dela Cuadra, 2005; - advanced)   Saxophone    • http://hyperphysics.phy-astr.gsu.edu/hbase/Music/sax.html  (outline)    • http://newt.phys.unsw.edu.au/jw/saxacoustics.html  (detailed discussion and resource database)    • https://syos.co/en-us/blogs/news/tagged/acoustics  (inside the saxophone)    • https://www.tutorhunt.com/resource/3462  (physics of saxophone)    • https://www.youtube.com/watch?v=nMjCwPDbFfU  (saxophone quintet performance - 1st part of Vivaldi's "Spring"; "Four Seasons" - video)   Cymbal    • https://sabian.com/cymbals-101/  (overview)    • https://www.youtube.com/watch?v=F2rxNwYcsN8  ("how it's made" - video)    • https://www.vsl.co.at/academy/percussion/cymbals & https://www.vsl.co.at/academy/percussion/suspended-cymbal  (outline)    • https://www.soundonsound.com/techniques/analysing-metallic-percussion  (detailed outline - Sound on Sound magazine)    • https://scholarship.rollins.edu/cgi/viewcontent.cgi?article=1020&context=stud_fac  (detailed exploration - cymbal vibration modes)    • https://iopscience.iop.org/article/10.1088/1742-6596/744/1/012110/pdf  (detailed exploration - striking effects)    • https://www.jstage.jst.go.jp/article/ast/22/3/22_3_177/_pdf  (acoustics of percussion instruments - T.D. Rosing, 2001)   Floor Tom    • http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cirmem.html  (circular membrane vibration modes)    • https://www.vsl.co.at/academy/percussion/field-drum  (tenor/field drum - no snares)    • http://www.bustedgear.com/faq_Drum_miking.html  (overview)    • http://drumzone.com.au/how_a_drum_works_9.html  (general overview of a drum kit)    • https://www.yamaha.com/en/musical_instrument_guide/drums  (extended overview of a drum kit)    • https://www.soundonsound.com/techniques/physics-percussion  (detailed outline - Sound on Sound magazine)    • https://www.jstage.jst.go.jp/article/ast/22/3/22_3_177/_pdf  (acoustics of percussion instruments - T.D. Rosing, 2001)

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