Fundamentals of Sound
MODULE 2B: BASIC ACOUSTICS (continued)
Resonance - Sound Propagation - Musical Instruments

 

 

Resonance (Resonators / Filters)
Sound Propagation
   Reflection - Diffusion - Reverberation - Absorption
   Diffraction - Refraction - The Doppler effect
Musical Instruments as Sound-Wave Generators and Transmitters

 


 

  

Resonance

 

In science, the term resonance describes the phenomenon of maximum energy transfer among two or more oscillating systems, occurring as the vibration frequency of the energy-supplying system (driving system) approaches the natural frequency (resonating frequency) of the receiving system. Matching the driving and natural frequencies maximizes resonance build-up of energy in the receiving system. (see this interactive applet - requires the Java plug-in)
Energy build-up from resonance can have extreme outcomes (e.g.   Tacoma Narrows Bridge collapse;    voice breaking glass).

Strings, drumheads, reeds, metal bars, and other sound producing parts of musical instruments, when excited, vibrate at their resonant/natural frequencies.
Resonators of musical instruments (e.g. body and air cavity of a guitar or a violin; air column/cone and material of a clarinet/saxophone/trumpet/organ pipes/vibraphone tubes/etc.) function as amplifiers, enhancing frequencies that match or are close to the instruments' range of natural frequency(ies). 

 
Natural frequency means 'the frequency with which a system would vibrate if energy was supplied to it and then it was left on its own.'
Think of pushing a swing, plucking a string, or striking a drum and then letting them vibrate on their own without further intervention.
 
Natural (resonant) frequencies of physical bodies depend on their size, shape, stiffness/tension, material/mass/density, etc.
 
General guidelines to determine natural frequencies of vibrating bodies:
  • the longer a string (e.g. on a guitar) or an air column (e.g. inside a trombone), the lower its natural frequency;
    in general, frequency is inversely proportional to length (for wooden/metal bars, frequency is inversely proportional to the square of the length)
     
  • the more massive a string or a drum head, or the larger the air volume inside a space/instrument, the lower its natural frequency;
    in general, frequency is inversely proportional to mass
     
  • the higher the tension on a string or a drum head, or the stiffer the vibrating material (e.g. xylophone/vibraphone bars), the higher its natural frequency;
    in general, frequency is proportional to the square root of tension/stiffness
     
  • the more symmetrical a shape and the less discontinuities, the simpler the resonance (i.e. the less frequency components in the spectrum of the resonance) and the more harmonic its spectrum (i.e. with frequencies that are equal or close-to-equal to integer multiples of the lowest natural frequency);
    For example
    a) a perfect cylinder whose air cavity is excited just by an air stream (flute; pan flutes, etc.) will produce a sound signal with a harmonic spectrum, dominated by its fundamental
    b) a struck cymbal will produce a sound signal with an inharmonic spectrum, dominated by higher frequency components (overtones).

Most musical instruments incorporate resonators with shapes, materials, and construction that result in a range of 'natural' frequencies so that they may respond to multiple frequency components of a single note and to more than one note.
In other words, musical resonators are, in general, broadly tuned (i.e. are shaped and constructed in ways that support resonance at, and therefore amplification of a wide/broad range of frequencies rather than a single frequency).
 


 

 
Conductive (forced) Resonance
 The resonator (e.g. guitar body) starts vibrating because it is in physical contact (e.g. via the bridge) with a vibrating body (e.g. strings) , which "forces" the resonator to replicate the oscillations
(animation example of conductive resonance, showing exaggerated modal vibrations of a guitar body in response to string vibrations)

Sympathetic (free) Resonance
The resonator (e.g. marimba's resonating tubes) starts vibrating in response to wave energy (e.g. from the vibration of struck wooden bars) reaching it via the air
(video example of sympathetic resonance)

 
An intermediate condition between
   a) full contact (direct, strong coupling; conductive resonance) and
   b) no contact (air coupling; sympathetic resonance)
is weak coupling/conductive resonance (e.g. multiple pendulums with their masses connected by springs).
Weak coupling can promote energy build-up through synchronization among multiple oscillators (metronome synchronization video example). Optional details here.

The figures, below, show the response (y axis) of a resonator with natural frequency f0 (=440Hz, on the second figure) when driven at various frequencies (x axis) and at different degrees of damping/friction/resistance/drain (in the figure, below-right, blue: low damping; green: medium damping; red: high damping).
 

Additional short video on the relationship between driver frequency and resonator response.


Observations on Resonator Response
 
Resonator response:

_ is highest when driven at or near its natural frequency
 
_ is greater when driven at frequencies that are below versus above natural frequency;

for driving frequencies below natural frequency, resonator response matches, at a minimum, the driver's amplitude
 
resonator response drops quicker as driving frequency extends above versus below natural frequency, dropping to almost zero

_ response level & sharpness and "best" resonance frequency decrease as damping increases
 
Generally, the broader the frequency response the lower the maximum response level (why?)
 

 
Resonators are analogous to filters.
Both, resonators and filters selectively amplify and/or attenuate different frequencies based on their (physical/electronic/digital) characteristics. The figures, below, illustrate the four most common types of idealized filters.
 
Resonator/Filter Damping; Bandwidth; Q factor

_ Q factor (Quality factor or 'sharpness') of a filter/resonator is a measure of its frequency response width. The heavier the damping, the broader the filter response (i.e. the wider the filter shape) and the lower its Q factor).

_ The lighter the damping, the narrower the filter response and the higher its Q factor.

_ Q = [Resonant Frequency] / [Response Frequency Bandwidth]

Frequency Bandwidth: upper minus lower frequency value of a frequency range

_ 3dB Response Bandwidth:
    frequency bandwidth across which response drops by 3dB

_ 10dB Response Bandwidth:
    frequency bandwidth across which response drops by 10dB

_ Filter slope:
    response level change over one octave (see figure to the right) 

Why, do you think, 3dB and 10dB are chosen as markers for a filter's shape?
 

 

The figures, below, are spectral envelopes, illustrating the frequency response of two popular vocal microphones, consequently referred to as frequency response curves.

They show:
a) how each microphone resonates  (mechanically, via its diaphragm, and electronically, via its circuit) in response to a sound wave and
b) how this response varies with frequency.
These graphs are spectra (they plot level over frequency) but only include the spectral envelope line (line connecting the tips of each implied, underlying spectral component).

The frequency response of an ideal microphone would be a flat horizontal line, parallel to the frequency axis, crossing the level axis at (or near) 0. This would mean that the microphone would respond equally well and efficiently at all frequencies and, consequently, would capture the original sound wave as is, without altering it.
Such response is not practically feasible (why?). The better the microphone, the closer it approximates it.
In addition, the flatter the frequency response, the less likely a microphone is to produce feedback (why?).
 

 

Resonance of stings and air columns
Standing waves are a form of resonance and develop at the natural frequency(ies) of oscillating systems.

The harmonic spectral components, or harmonics, of the tones produced by standing waves on a string (below) correspond to wavelengths λ that relate directly to the length of the string L.
For each standing-wave pattern or mode, each loop corresponds to 1/2 wavelength. So, the 1st (fundamental), 2nd, and 3rd harmonics have the following wavelengths with respect to string length: λ1=2L;    λ2=1L;    λ3=2/3L.

The two resources, to the right, illustrate the generation of sawtooth standing waves on strings, initiated through conductive (i.e. forced) resonance

 
Analogously, the harmonic spectral components, or harmonics, of the tones produced by standing waves / modes within an air column (below) also correspond to wavelengths λ that relate directly to the length of the air column L. Again, for each standing wave pattern, each loop corresponds to 1/2 wavelength (idealized contexts/shapes).

Cylinder open at both ends (left image - flutes, pan flutes, some organ pipes, etc.):
The 1st (fundamental), 2nd, 3rd, and 4th harmonics have the following respective wavelengths:  λ1=2L;   λ2=1L;   λ3=2/3L λ4=1/2L
Cylinder closed at one end
(middle image - clarinet, several non western woodwinds, etc.):
The 1st, 2nd, 3rd, and 4th harmonics have the following respective wavelengths: λ1=4L;   λ2=--;   λ3=4/3L;   λ4=-- (only odd components)
Cone closed at one end (right image - oboe, saxophone, etc.):
The 1st, 2nd, 3rd, and 4th harmonics have the following respective wavelengths: λ1=2L;   λ2=1L;   λ3=2/3L λ4=1/2L


 
The full story is less simple, complicated by tension changes, as a string is excited, and by effective air-column length changes, as a column's diameter and shape change.

For most practical purposes, and assuming the same length, the lowest note produced by conical instruments (e.g. saxophone), cylindrical instruments open on both ends (e.g. flute), and strings, has similar wavelength/frequency across instruments. The lowest note produced by cylindrical instruments of the same length, closed on one end (e.g. clarinet), has wavelength/frequency corresponding to a note one octave lower. 
 

 
Resonances of metal plates (e.g. cymbals) produce signals with inharmonic spectra (i.e. with components whose frequencies are not integer multiples of the lowest frequency) and tones with no definite pitch.
The relationship between cymbal dimensions and wavelengths is quite complex but cymbal vibrations do follow specific vibration patterns referred to as modes. (see here and below).

 

Helmholtz resonators - Bass reflex loudspeakers
 
Helmholtz resonators, named after their inventor (1800s), are spherical or cylindrical containers with a short, narrow neck, open at one end. 
The air in the container acts like a spring and the air in the neck acts like a mass, vibrating because of the 'springiness' of the air inside.
 
Consider a 'lump' of air at the neck of the bottle (shaded in the middle diagrams). 

_ An air jet can force this lump of air a little way down the neck, compressing the air inside. 
_ The increased pressure of the compressed air, inside, drives the 'lump' of air out but, when it gets to its original position, its momentum moves it a small distance outside the neck. 
_ This rarifies the air inside the body, which then sucks the 'lump' of air back in.  Therefore, the 'lump' of air in the neck and the air inside the body behave like a mass/spring system.

A jet of air from your lips, for example, is capable of deflecting alternately into the bottle and outside, and that provides the power to keep the oscillation going.
The frequency of the vibration is related to the speed of sound in air c, the length l and area A of the container neck, and the volume V of the container. 
As indicated by the equation, above, longer necks ( l ) and larger containers (V) produce lower frequencies while wider necks (A) produce higher frequencies.
 
An example of the application of Helmholtz resonance is the inclusion of a bass reflex enclosure (or tuned port) to a loudspeaker
The bass reflex enclosure improves the bass (low) frequency efficiency and range of a loudspeaker by carefully adjusting the shape and position of an opening/tube connecting the inside of the speaker box with the outside. The air volume of the box acts as the air in the body of a Helmholtz resonator, with a resonant frequency that is determined by the geometry of the hole/tube and enclosure, deliberately chosen to smoothly extend the frequency range of the speaker system below its original low cutoff frequency.
In addition, the existence of the port greatly reduces the air pressure variation between the inside and the outside of the speaker box facilitating transmission of wave energy from the inside of the instrument to the outside (see reflection / transmission, below).

 


SOUND PROPAGATION

Reflection - Transmission - Diffusion/Reverberation - Absorption

 

When a sound wave meets a boundary separating two media (1 & 2), some of the wave energy may be

reflected (bounce) back into medium 1;
     
The energy that will be reflected may generate standing waves, echoes, or reverberation.
transmitted
(pass through) into medium 2;
     
The energy that will pass through/penetrate the boundary may be refracted (bend) inside, absorbed by, and/or simply transmitted into medium 2.
diffracted
(bend around the boundary).

Reflection

Acoustics borrows the basic law of reflection from optics: The angle of incidence will be equal to the angle of reflection.

 

The balance between reflected and transmitted energy at a boundary is determined by the impedance (resistance) relationship of the areas either side of the boundary:
The larger the impedance mismatch and the larger the angle of incidence the greater the reflected energy.
Practically all musical instruments rely on reflection to build up standing waves and produce sound. They achieve this by involving structures with appropriately placed boundaries and impedance mismatches.
 
If

z1: impedance of the medium from where the sound wave originates
z2: impedance of the medium to which the sound wave moves

then

If z1 = z2:  no reflection / total transmission
If z2 < z1:  reflected & incident waves in phase
If z2 > z1:  reflected & incident waves 1800 out of phase (below, right)
If z2 <<< z1 or z2 >>> z1:  total reflection / no transmission (below, left) 


OPTIONAL: More specifically,
If  
I0
: Incident Intensity   Ir: Reflected intensity  It: Transmitted intensity
then 
Proportion of Reflected Intensity   = Ir / I0 = [(z1 / z2) -1 / (z1 / z2) + 1]2   =   [(z2-z1) / (z2+z1)]2  
      (for a detailed derivation of the above see here - then, remember that intensity is proportional to the square of pressure)
Proportion of Transmitted Intensity = It / I0 = 1 - Ir / I0 = 1 -  [(z2-z1) / (z2+z1)]2  = 4z2z1 / (z2+z1)2
 
 


 
Examples of reflection - dependence on impedance
(from the University of Saskatchewan, Department of Engineering Physics)

Total reflection at a free end (Z2 = 0)

Z1/Z2 = 2, (i.e. the second medium is lighter)
(reflected plus transmitted energy = original energy)

Total reflection at a fixed end (Z2 = infinity)

Z1/Z2 = 0.5 (i.e. the second medium is heavier)
The reflected wave reverses its phase
(reflected plus transmitted energy = original energy)

 

Successive, partial, uncorrelated (not coherent / not in the same or symmetrical direction) reflections from the surfaces in an enclosure, such as the uneven complex-shaped walls of an auditorium, result in the scattering of the sound waves and diffusion of sonic energy (see below). Repeated, diffused reflections are perceived as reverberation
 
Reverberation is a desirable property of auditoriums to the extent that it
a)
helps overcome the inverse square law drop-off of sound intensity in the enclosure and
b)
contributes to the sonic character of a musical style.

 


 
Reverberation time is defined as the time it takes for the sound resulting from multiple diffuse reflections to lose 60dB of its original level. 
Explore qualitatively these two pages: Reverberation & Reverberation Time (Hyperphysics, Department of Physics and Astronomy, Georgia State University). You are not expected to follow the links within them or do any calculations.
More on reverberation in the following weeks.
 
Absorption describes the loss of energy as a wave propagates through a medium and/or strikes a boundary (see to the left).

For example, sound can be absorbed through

a) friction as a sound wave propagates in air,
b) trapping as a sound wave strikes a pliable and/or porous boundary,
c) thermal loss of energy, or
d) decoupling and interference in layered boundaries.

The degree to which a medium absorbs sound is expressed by its coefficient of absorption, a.
Absorption coefficients range from 0 to 1, are related to the medium's material and structure, and are always frequency-dependent.
 
See here for a brief qualitative outline of absorption and for a table of absorption coefficients of common materials.
Additional table with absorption coefficients.

 


Diffraction - Refraction - The Doppler Effect

 

Diffraction
 
Diffraction describes the ability of sound waves to bend around obstacles and through openings whose smallest dimension is smaller than the wavelength(s) of the sound waves. Diffraction is very important in that it largely determines the radiation patterns of (i.e. the ways the sound spreads out of) sound sources and the pick-up patterns of (i.e. the ways the sound comes into) microphones. 

In general, low frequencies diffract more than high frequencies, as it is more likely for obstacles/openings commonly encountered in everyday life to be smaller than the wavelength (λ) of low, rather than high frequencies. More specifically, diffracted energy begins to be appreciable at wavelengths ≈ half the size of the opening /obstacle in question and becomes almost total at wavelengths four times the opening/obstacle.
Play around with this simulation to explore the impact of opening size to diffraction (select the "slits" option and experiment with the frequency and slit width faders).

Example wavelengths, assuming c = 345m/s (210C)
a) f = 66Hz (e.g. fundamental freq. of ~ a C2 note played on a bass guitar) => T = 1/66 = 0.01515s;   λ = c*T = 345*0.01515 => λ=5.23m/cycle   (alternatively, λ = c/f = 345/66 => λ=5.23m/cycle)
a) f = 528Hz (e.g. fundamental freq. of ~ a C5 note played on a flute) => T = 1/528 = 0.00189s;  λ = c*T = 345*0.00189 => λ=0.65m/cycle   (alternatively, λ = c/f = 345/528 => λ=0.65m/cycle)

Most sounds correspond to complex signals and have spectra with more than one frequency component. Consequently, portions of the sound's spectrum may be diffracted and portions may not, resulting in a perceived sound quality that depends on the position of a listener relative to the "sound-source - obstacle/opening" system in question. More during the discussion on timbre.
 

Illustrations of diffraction - dependence on frequency
(from Hyperphysics, GSU)

   
 

Refraction describes a change in the propagation direction of sound waves as they move across two media where sound waves travel at different longitudinal velocities. See Refraction (Hyperphysics, GSU)
       Optional: According to Snell's law, sinθ1 / sinθ2 = v1 / v2
                        Similarly to the law of reflection, Snell's law can also be derived from Fermat's principle.

 
In the figure to the right, v2 < v1 and the sound waves refract/bend towards the "normal" (:dotted line - line perpendicular to the boundary).
The sound waves would bend in the opposite direction if v2 > v1

Since the speed of sound in air depends on temperature, sound waves may be refracted as they cross layers of air that are at different temperatures (e.g. as they rise above the ground). Wind direction and speed also influences the propagation velocity of sound waves and, consequently, refraction angle. 
Therefore, orientation of an open air theater stage relative to prevailing winds in the area, as well as air temperature considerations are crucial to efficient sound transmission to the audience and minimization of sound disturbance to the surrounding areas.

Play around with this interactive applet exploring the phenomenon of refraction.

The figures, below, illustrate the refraction effects of temperature and wind differentials.

 

     

 
Optional: Both, refraction and diffraction are theoretically explained through
a combination of the Huygens-Fresnel and the linear superimposition principles.
 

 

The Doppler effect
 

When a sound source moves relative to a listener, the emitted frequency will appear higher (as the source/listener approaches) or lower (as the source/listener recedes) than the emitted frequency.
The effect is directly linked to the relative speed between sound-source and listener. In short, wave fronts are compressed in-front of a moving source and are spread apart behind it.
Read this page for a multimedia-rich explanation.
For the Hyperphysics resource on the effect, click here (approach any equations qualitatively).
 
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.

A popular musical application of the Doppler effect is in the Leslie Organ Loudspeaker, producing a distinct frequency modulation (vibrato) effect (why?).
Applications of the Doppler effect (named after the Austrian physicist who first described it quantitatively in the 1800s) abound in optics, astronomy, medicine, audio, and much more (Optional: see here).
 

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

 


Explore select interactive tools in this resource (focus on the first three sections):
   _ Oscillations and Waves: Ripple Tank
   _ Acoustics: (all available)
   _ Signal Processing: Fourier Series

Additional Physics Simulations here

Bookmark and return to the following online resources to expand your understanding on any specific topic. Deep links on specific concepts within these resources are dispersed throughout our course site:
_ Hyperphysics: Physics and Psychophysics of Sound and Hearing
   (Department of Physics & Astronomy, Georgia State University, Atlanta, USA)
_ Music Acoustics
   (University of New South Whales, Sydney, Australia)
_ Tutorials and Interactive Labs on Sound, Waves, & Music
   (Online Physics Classroom)
_ Fundamentals of Musical Acoustics Video Playlist


 

 

Musical Instruments as Sound Generators and Transmitters

  

HISTORICAL APPROACH

 
The most common (but highly inconsistent) classification of musical instruments splits them into three categories:

Percussion
Winds
(brass & woodwind), and
Strings

The standard classification system of musical instruments originates in the early 1800s work by Mahillon, a curator of musical instruments at the Brussels Music Conservatory.
It was modified and expanded ~75 years later by Hornbostel and Sachs (optional resource with more details: the Hornbostel-Sacks system) to encompass instruments from cultures outside the Western musical tradition.

The system includes 5 principal categories, mainly (but not always) based on the material that is excited first when an instrument is played, and several subcategory levels, adding up to over 300 basic groupings:
 

Idiophones produce sound mainly through vibrations of their body - through 3-dimentional standing waves on the entire instrument (cymbals, xylophone, cowbell, etc.)
Membranophones produce sound mainly through vibrations of a stretched membrane - through 2-dimensional standing waves on membranes (all drums, kazoos, etc.)
Chordophones produce sound mainly through vibrations of one or more stretched strings - through 1-dimentional standing waves on strings (guitar, harp, piano, violin, etc.)
Aerophones produce sound mainly through vibrations of air within a volume outlined by the instrument - through 3-dimentional standing waves on bounded air volume(s) (saxophone, oboe, trumpet, organ, etc.)
Electrophones produce sound mainly through vibrations/oscillations of electrical voltage/current (synthesizers, theremin (explanation), etc.)

 
ACOUSTICS APPROACH
 

Within the context of acoustics, musical instruments are examined systematically in terms of (outlined in Hajda, J.M. (2009). Psychoacoustics and Organology. Journal of the Acoustical Society of America, 126: 2236):

  1. their physical properties
  2. the way energy is supplied, modified, and radiated, and
  3. the resulting sonic perceptions

We will, therefore, approach musical instruments as

  • a) driver – b) generator – c) resonator systems,
    with some sort of

    • coupling, facilitating energy transfer within the instrument (e.g. various bridges) and from the instrument to the surrounding air (e.g. various surfaces or openings),
      and with different types of

    • articulation, controlling how driver vibrations are initiated (e.g. by striking, blowing, plucking), "stopped" (i.e. how they change pitch along a pitch scale), maintained (e.g. by bowing, blowing), and end (e.g. with/without performer intervention). 

Voice acoustics has adopted a slightly different terminology for the stages of this analytical approach: 
a) actuator (driver) - b) vibrator (generator) - c) resonator - d) articulator(s)
Beyond the mere terminology differences in the first two items, the key difference is that articulators are included as the last stage of the voice analysis system because, in voice production, articulation occurs at the end of the sound production process and also facilitates the coupling of energy out the "instrument" and into the surrounding air.
For most other instruments, articulation can occur at any of the system's stages.
 

The heart of any musical instrument is its sound generator

This is the part that, when excited by a driving force, will be the first to vibrate, initiating the sound waves that travel through the instrument to the surrounding air and eventually reach our ears or a microphone.
Some examples of generators include: strings (violin, piano,...), reeds (clarinet, sax,...), air (flutes,...), lips (trumpet,...), skins/membranes (various drums,...), metal plates (cymbals, gongs,…), vocal folds (voice). 
The vibration characteristics of a generator outline the frequency and spectral ranges of the sounds produced.

_ Some generators are variable. For example, strings or air columns can be "stopped" (e.g. by pressing the string over a fret, pushing a trumpet valve, or gliding the outer slide tube of a trombone), changing their length and, therefore, their resonant characteristics.
_ Some generators are fixed. For example, any aerophone with no openings, valves, or sliding mechanism on their body (e.g. natural horn; bugle), the piano strings, or various percussion.

The driver that excites (sets into vibration) a generator can apply or articulate a driving force:

  • in various ways: e.g. instantaneous (plucking, striking, etc.) or continuous (bowing, blowing, etc.).
    Note: Instantaneous application of a driving force produces signals whose attack portion is the most salient.

    • for instantaneous application: at various contact durations: e.g. plucking a string with the fingertips or a plectrum; striking a drum with a felt-tip or a wooden-tip mallet (i.e. longer vs. shorter contact).
      Note: The faster the rate and the shorter the contact, the more complex and important (perceptually) the attack portion of a signal's envelope to the resulting sound and the richer in high frequency components the resulting spectrum.
       

    • for continuous application: with various degrees of modulation: e.g. vibrato (narrow frequency/pitch modulation or fast trills), tremolo (narrow amplitude/loudness modulation or fast repetitions of the same note), portamento (continuous slide up/down in frequency/pitch), or glissando (discrete move up/down in frequency/pitch).
      Note: The more time-variance in a signal the more salient it becomes in the presence of other signals (i.e. it can easier cut through other simultaneous signals).
       

  • at various locations on the generator: e.g. in the middle (i.e. away from any support); towards the edges (i.e. near a support).
    Note: The closer a driving force is applied to the middle of a generator (such as a string, a membrane, or a bar) the richer in low frequency components the resulting spectrum.

In general, the amount of driving force outlines the dynamic range of the sounds produced and the manner in which it is applied modifies the signal and spectral envelopes of the sounds produced.
 

A generator is coupled to (connected to / placed near), and therefore amplified, sustained, and “shaped” (in terms of spectrum) by some sort of resonating system

A musical instrument's resonating system defines the system's frequency response: the degree to which energy provided by the generator is amplified and/or reduced within a frequency range of interest.
Examples of resonating systems include: soundboard (piano,...), air cavity & body (lutes, violins, drums, thumb piano (mbira/kalimba...), air column (horns,...), vocal tract (voice), mouth cavity (jew’s-harp,...), etc. 

Size, shape, material, and construction of a resonator are all parameters that determine its resonance characteristics and its influence on the sound produced. 
Resonating systems incorporated to musical instruments are preferably broadly (rather than sharply) tuned and have an even frequency response over a broad range of frequencies.  In other words, they are designed in such a way that they can more-or-less evenly resonate 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.
 

A coupler provides the link between generator and resonator (e.g. various types of bridges etc.), and mainly modifies their degree of impedance matching/mismatching.  In the case of coupling through proximity, the coupler is air. 

There is always some degree of energy feedback between generators and resonators (that depends on coupling and may be different for different notes on the same instrument), meaning that generators and resonators may alternate roles and sometimes do so in a periodic/regular manner.
The degree of feedback and 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 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 (the way sound waves are transferred from the instrument to the surrounding air; e.g. the bell on a horn) influences the spectral characteristics of the sound leaving the instrument, as well as its radiation pattern (see below.)
 

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.
 
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).

The radiation efficiency and radiation pattern of a musical instrument play a decisive role in the relationship between a) the position of a listener or microphone relative to the instrument and b) its perceived timbre.
 

 
Resources on example instruments
  
 
Standard online resources addressing multiple instruments include:
Yamaha's Music Instrument Guides
Vienna Symphonic Library's Instumentology

Hyperphysics' section on Musical Instruments
Music Acoustics site, University of South Wales
The Physics of Music and Musical Instruments by David R. Lapp (Tufts University)

• the relevant Wikipedia pages

Voice
http://hyperphysics.phy-astr.gsu.edu/hbase/Music/voice.html
     (outline)
http://newt.phys.unsw.edu.au/jw/voice.html
     (only read the "Introduction and overview" section up to, and including, the "Source-filter interactions" subsection)
https://voicefoundation.org/health-science/voice-disorders
     (resource on voice production, health, and conservation)
http://www.music.mcgill.ca/~gary/courses/papers/
Sundberg_SingingVoice_ScientificAmerican_1977.pdf

     (seminal 1977 paper by pioneer in voice acoustics, Johan Sundberg)
https://www.youtube.com/watch?v=ZLgAQTMgZ6g
     (voice production acoustics - video)
https://muse.union.edu/griffiths-capstone
     (Modeling the Human Vocal Tract - student project)
https://youtu.be/UkNDBQmEMSY
     (formants / more advanced; with download link to a voice synth, based on formant manipulation - video)

Guitar
http://hyperphysics.phy-astr.gsu.edu/hbase/Music/guita.html
     (outline)
http://newt.phys.unsw.edu.au/music/guitar
     (only read "Basics about sound ...", "Anatomy of an acoustic....", and "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
     (overview)
https://www.vsl.info/en/academy/percussion/marimba
     (construction; sound production; etc. - Vienna Symphonic Library)
https://www.youtube.com/watch?v=pCnFE-lSta4  &  https://www.youtube.com/watch?v=Bsj-xx25dHY 
     (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)

Trombone
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.pdff
     (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
     (another outline)
http://newt.phys.unsw.edu.au/music/flute
     (detailed overview and resource database)
http://www.flutepage.de/deutsch/goodies/physik.php?englisch=true
     (detailed overview)
https://ccrma.stanford.edu/~pdelac/research/MyPublishedPapers/Thesis.pdf
     (doctoral dissertation - Stanford University - beyond the scope of this course)
https://www.youtube.com/watch?v=Ho5Fc62kBJg
     (acoustics of flute techniques; performance - video)

Saxophone
http://hyperphysics.phy-astr.gsu.edu/hbase/Music/sax.html
     (outline)
http://newt.phys.unsw.edu.au/jw/saxacoustics.html
     (detailed overview 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 - video)

Cymbal
https://sabian.com/cymbals-101/
     (overview)
https://www.vsl.co.at/en/Percussions/Suspended_cymbal & https://www.vsl.co.at/en/Percussions/Cymbals
     (outline - Vienna Symphonic Library)
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://youtu.be/kpoanOlb3-w  -  https://www.youtube.com/watch?v=F2rxNwYcsN8
     (cymbal vibration and 'how it's made" videos)
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/en/Percussions/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)
 

 


 

  

Loyola Marymount University - School of Film & Television