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Friday, July 3, 2015

Harmonic oscillator

From Wikipedia, the free encyclopedia
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x:
\vec F = -k \vec x \,
where k is a positive constant.

If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).

If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:
  • Oscillate with a frequency lower than in the non-damped case, and an amplitude decreasing with time (underdamped oscillator).
  • Decay to the equilibrium position, without oscillations (overdamped oscillator).
The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called "critically damped."

If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator.

Mechanical examples include pendulums (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.

Simple harmonic oscillator

Simple harmonic motion

A simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists of a mass m, which experiences a single force, F, which pulls the mass in the direction of the point x=0 and depends only on the mass's position x and a constant k. Balance of forces (Newton's second law) for the system is
F = m a = m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = m\ddot{x} = -k x.
Solving this differential equation, we find that the motion is described by the function
x(t) = A\cos\left( \omega t+\phi\right),
where
\omega = \sqrt{\frac{k}{m}} = \frac{2\pi}{T}.
The motion is periodic, repeating itself in a sinusoidal fashion with constant amplitude, A. In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period T, the time for a single oscillation or its frequency f = 1⁄T, the number of cycles per unit time. The position at a given time t also depends on the phase, φ, which determines the starting point on the sine wave. The period and frequency are determined by the size of the mass m and the force constant k, while the amplitude and phase are determined by the starting position and velocity.

The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position but with shifted phases. The velocity is maximum for zero displacement, while the acceleration is in the opposite direction as the displacement.

The potential energy stored in a simple harmonic oscillator at position x is
U = \frac{1}{2}kx^2.

Damped harmonic oscillator

Dependence of the system behavior on the value of the damping ratio ζ
A damped harmonic oscillator, which slows down due to friction

Another damped harmonic oscillator

In real oscillators, friction, or damping, slows the motion of the system. Due to frictional force, the velocity decreases in proportion to the acting frictional force. While simple harmonic motion oscillates with only the restoring force acting on the system, damped harmonic motion experiences friction. In many vibrating systems the frictional force Ff can be modeled as being proportional to the velocity v of the object: Ff = −cv, where c is called the viscous damping coefficient.

Balance of forces (Newton's second law) for damped harmonic oscillators is then
F = F_{ext} - kx - c\frac{\mathrm{d}x}{\mathrm{d}t} = m \frac{\mathrm{d}^2x}{\mathrm{d}t^2}.
When no external forces are present (i.e. when F_{ext}=0), this can be rewritten into the form
\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^{\,2} x = 0,
where
\omega_0 = \sqrt{\frac{k}{m}} is called the 'undamped angular frequency of the oscillator' and
\zeta = \frac{c}{2 \sqrt{mk}} is called the 'damping ratio'.

Step-response of a damped harmonic oscillator; curves are plotted for three values of μ = ω1 = ω0√1−ζ2. Time is in units of the decay time τ = 1/(ζω0).

The value of the damping ratio ζ critically determines the behavior of the system. A damped harmonic oscillator can be:
  • Overdamped (ζ > 1): The system returns (exponentially decays) to steady state without oscillating. Larger values of the damping ratio ζ return to equilibrium slower.
  • Critically damped (ζ = 1): The system returns to steady state as quickly as possible without oscillating (although overshoot can occur). This is often desired for the damping of systems such as doors.
  • Underdamped (ζ < 1): The system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero. The angular frequency of the underdamped harmonic oscillator is given by
\omega_1 = \omega_0\sqrt{1 - \zeta^2}.
The Q factor of a damped oscillator is defined as
Q = 2\pi \times \frac{\text{Energy stored}}{\text{Energy lost per cycle}}.
Q is related to the damping ratio by the equation Q = \frac{1}{2\zeta}.

Driven harmonic oscillators

Driven harmonic oscillators are damped oscillators further affected by an externally applied force F(t).

Newton's second law takes the form
F(t)-kx-c\frac{\mathrm{d}x}{\mathrm{d}t}=m\frac{\mathrm{d}^2x}{\mathrm{d}t^2}.
It is usually rewritten into the form
\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{F(t)}{m}.
This equation can be solved exactly for any driving force, using the solutions z(t) which satisfy the unforced equation:
\frac{\mathrm{d}^2z}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}z}{\mathrm{d}t} + \omega_0^2 z = 0,
and which can be expressed as damped sinusoidal oscillations,
z(t) = A \mathrm{e}^{-\zeta \omega_0 t} \ \sin \left( \sqrt{1-\zeta^2} \ \omega_0 t + \phi \right),
in the case where ζ ≤ 1. The amplitude A and phase φ determine the behavior needed to match the initial conditions.

Step input

In the case ζ < 1 and a unit step input with x(0) = 0:
{F(t) \over m} = \begin{cases} \omega _0^2 & t \geq 0 \\ 0 & t < 0 \end{cases}
the solution is:
x(t) = 1 - \mathrm{e}^{-\zeta \omega_0 t} \frac{\sin \left( \sqrt{1-\zeta^2} \ \omega_0 t + \varphi \right)}{\sin(\varphi)},
with phase φ given by
\cos \varphi = \zeta. \,
The time an oscillator needs to adapt to changed external conditions is of the order τ = 1/(ζω0). In physics, the adaptation is called relaxation, and τ is called the relaxation time.

In electrical engineering, a multiple of τ is called the settling time, i.e. the time necessary to ensure the signal is within a fixed departure from final value, typically within 10%. The term overshoot refers to the extent the maximum response exceeds final value, and undershoot refers to the extent the response falls below final value for times following the maximum response.

Sinusoidal driving force


Steady state variation of amplitude with relative frequency \omega/\omega_0 and damping \zeta of a driven simple harmonic oscillator.

In the case of a sinusoidal driving force:
\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{1}{m} F_0 \sin(\omega t),
where \,\!F_0 is the driving amplitude and \,\!\omega is the driving frequency for a sinusoidal driving mechanism. This type of system appears in AC driven RLC circuits (resistor-inductor-capacitor) and driven spring systems having internal mechanical resistance or external air resistance.

The general solution is a sum of a transient solution that depends on initial conditions, and a steady state that is independent of initial conditions and depends only on the driving amplitude \,\!F_0, driving frequency, \,\!\omega, undamped angular frequency \,\!\omega_0, and the damping ratio \,\!\zeta.

The steady-state solution is proportional to the driving force with an induced phase change of \,\!\phi:
x(t) = \frac{F_0}{m Z_m \omega} \sin(\omega t + \phi)
where
Z_m = \sqrt{\left(2\omega_0\zeta\right)^2 + \frac{1}{\omega^2}\left(\omega_0^2 - \omega^2\right)^2}
is the absolute value of the impedance or linear response function and
\phi = \arctan\left(\frac{2\omega \omega_0\zeta}{\omega^2-\omega_0^2 }\right)
is the phase of the oscillation relative to the driving force, if the arctan value is taken to be between -180 degrees and 0 (that is, it represents a phase lag, for both positive and negative values of the arctan's argument).

For a particular driving frequency called the resonance, or resonant frequency \,\!\omega_r = \omega_0\sqrt{1-2\zeta^2}, the amplitude (for a given \,\!F_0) is maximum. This resonance effect only occurs when \,\zeta < 1 / \sqrt{2}, i.e. for significantly underdamped systems. For strongly underdamped systems the value of the amplitude can become quite large near the resonance frequency.

The transient solutions are the same as the unforced (\,\!F_0 = 0) damped harmonic oscillator and represent the systems response to other events that occurred previously. The transient solutions typically die out rapidly enough that they can be ignored.

Parametric oscillators

A parametric oscillator is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of the oscillator, such as the damping or restoring force. A familiar example of parametric oscillation is "pumping" on a playground swing.[1][2][3] A person on a moving swing can increase the amplitude of the swing's oscillations without any external drive force (pushes) being applied, by changing the moment of inertia of the swing by rocking back and forth ("pumping") or alternately standing and squatting, in rhythm with the swing's oscillations. 
The varying of the parameters drives the system. Examples of parameters that may be varied are its resonance frequency \omega and damping \beta.
Parametric oscillators are used in many applications. The classical varactor parametric oscillator oscillates when the diode's capacitance is varied periodically. The circuit that varies the diode's capacitance is called the "pump" or "driver". In microwave electronics, waveguide/YAG based parametric oscillators operate in the same fashion. The designer varies a parameter periodically to induce oscillations.

Parametric oscillators have been developed as low-noise amplifiers, especially in the radio and microwave frequency range. Thermal noise is minimal, since a reactance (not a resistance) is varied. Another common use is frequency conversion, e.g., conversion from audio to radio frequencies. For example, the Optical parametric oscillator converts an input laser wave into two output waves of lower frequency (\omega_s, \omega_i).

Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. Parametric excitation differs from forcing, since the action appears as a time varying modification on a system parameter. This effect is different from regular resonance because it exhibits the instability phenomenon.

Universal oscillator equation

The equation
\frac{\mathrm{d}^2q}{\mathrm{d} \tau^2} + 2 \zeta \frac{\mathrm{d}q}{\mathrm{d}\tau} + q = 0
is known as the universal oscillator equation since all second order linear oscillatory systems can be reduced to this form. This is done through nondimensionalization.

If the forcing function is f(t) = cos(ωt) = cos(ωtcτ) = cos(ωτ), where ω = ωtc, the equation becomes
\frac{\mathrm{d}^2q}{\mathrm{d} \tau^2} + 2 \zeta \frac{\mathrm{d}q}{\mathrm{d}\tau} + q = \cos(\omega \tau).
The solution to this differential equation contains two parts, the "transient" and the "steady state".

Transient solution

The solution based on solving the ordinary differential equation is for arbitrary constants c1 and c2
q_t (\tau) = \begin{cases} \mathrm{e}^{-\zeta\tau} \left( c_1 \mathrm{e}^{\tau \sqrt{\zeta^2 - 1}} + c_2 \mathrm{e}^{- \tau \sqrt{\zeta^2 - 1}} \right) & \zeta > 1 \text{ (overdamping)} \\ \mathrm{e}^{-\zeta\tau} (c_1+c_2 \tau) = \mathrm{e}^{-\tau}(c_1+c_2 \tau) & \zeta = 1 \text{ (critical damping)} \\ \mathrm{e}^{-\zeta \tau} \left[ c_1 \cos \left(\sqrt{1-\zeta^2} \tau\right) +c_2 \sin\left(\sqrt{1-\zeta^2} \tau\right) \right] & \zeta < 1 \text{(underdamping)} \end{cases}

The transient solution is independent of the forcing function.

Steady-state solution

Apply the "complex variables method" by solving the auxiliary equation below and then finding the real part of its solution:
\frac{\mathrm{d}^2 q}{\mathrm{d}\tau^2} + 2 \zeta \frac{\mathrm{d}q}{\mathrm{d}\tau} + q = \cos(\omega \tau) + \mathrm{i}\sin(\omega \tau) = \mathrm{e}^{ \mathrm{i} \omega \tau} .
Supposing the solution is of the form
\,\! q_s(\tau) = A \mathrm{e}^{\mathrm{i} ( \omega \tau + \phi ) } .
Its derivatives from zero to 2nd order are
q_s = A \mathrm{e}^{\mathrm{i} ( \omega \tau + \phi ) }, \ \frac{\mathrm{d}q_s}{\mathrm{d} \tau} = \mathrm{i} \omega A \mathrm{e}^{\mathrm{i} ( \omega \tau + \phi ) }, \ \frac{\mathrm{d}^2 q_s}{\mathrm{d} \tau^2} = - \omega^2 A \mathrm{e}^{\mathrm{i} ( \omega \tau + \phi ) } .
Substituting these quantities into the differential equation gives
\,\! -\omega^2 A \mathrm{e}^{\mathrm{i} (\omega \tau + \phi)} + 2 \zeta \mathrm{i} \omega A \mathrm{e}^{\mathrm{i}(\omega \tau + \phi)} + A \mathrm{e}^{\mathrm{i}(\omega \tau + \phi)} = (-\omega^2 A \, + \, 2 \zeta \mathrm{i} \omega A \, + \, A) \mathrm{e}^{\mathrm{i} (\omega \tau + \phi)} = \mathrm{e}^{\mathrm{i} \omega \tau} .
Dividing by the exponential term on the left results in
\,\! -\omega^2 A + 2 \zeta \mathrm{i} \omega A + A = \mathrm{e}^{-\mathrm{i} \phi} = \cos\phi - \mathrm{i} \sin\phi .
Equating the real and imaginary parts results in two independent equations
A (1-\omega^2)=\cos\phi \qquad 2 \zeta \omega A = - \sin\phi.

Amplitude part


Bode plot of the frequency response of an ideal harmonic oscillator.

Squaring both equations and adding them together gives
\left . \begin{array}{rcl} A^2 (1-\omega^2)^2 & = & \cos^2\phi \\[6pt] (2 \zeta \omega A)^2 & = & \sin^2\phi \end{array} \right \} \Rightarrow A^2[(1-\omega^2)^2 + (2 \zeta \omega)^2] = 1.
Therefore,
A = A( \zeta, \omega) = \text{sign} \left( \frac{-\sin\phi}{2 \zeta \omega} \right) \frac{1}{\sqrt{(1-\omega^2)^2 + (2 \zeta \omega)^2}}.
Compare this result with the theory section on resonance, as well as the "magnitude part" of the RLC circuit. This amplitude function is particularly important in the analysis and understanding of the frequency response of second-order systems.

Phase part

To solve for φ, divide both equations to get
\tan\phi = - \frac{2 \zeta \omega}{ 1 - \omega^2} = \frac{2 \zeta \omega}{\omega^2 - 1} \Rightarrow \phi \equiv \phi(\zeta, \omega) = \arctan \left( \frac{2 \zeta \omega}{\omega^2 - 1} \right ).
This phase function is particularly important in the analysis and understanding of the frequency response of second-order systems.

Full solution

Combining the amplitude and phase portions results in the steady-state solution
\,\! q_s (\tau) = A(\zeta,\omega) \cos(\omega \tau + \phi(\zeta,\omega)) = A\cos(\omega \tau + \phi).
The solution of original universal oscillator equation is a superposition (sum) of the transient and steady-state solutions
\,\! q(\tau) = q_t (\tau) + q_s (\tau).
For a more complete description of how to solve the above equation, see linear ODEs with constant coefficients.

Equivalent systems

Harmonic oscillators occurring in a number of areas of engineering are equivalent in the sense that their mathematical models are identical (see universal oscillator equation above). Below is a table showing analogous quantities in four harmonic oscillator systems in mechanics and electronics. If analogous parameters on the same line in the table are given numerically equal values, the behavior of the oscillators—their output waveform, resonant frequency, damping factor, etc.—are the same.

Translational Mechanical Torsional Mechanical Series RLC Circuit Parallel RLC Circuit
Position x\, Angle \theta\,\! Charge q\, Flux linkage \phi\,
Velocity \frac{\mathrm{d}x}{\mathrm{d}t}\, Angular velocity \frac{\mathrm{d}\theta}{\mathrm{d}t}\, Current \frac{\mathrm{d}q}{\mathrm{d}t}\, Voltage \frac{\mathrm{d}\phi}{\mathrm{d}t}\,
Mass M\, Moment of inertia I\, Inductance L\, Capacitance C\,
Spring constant K\, Torsion constant \mu\, Elastance 1/C\, Susceptance 1/L\,
Damping \gamma\, Rotational friction \Gamma\, Resistance R\, Conductance G=1/R\,
Drive force F(t)\, Drive torque \tau(t)\, Voltage e\, Current i\,
Undamped resonant frequency f_n\,:
\frac{1}{2\pi}\sqrt{\frac{K}{M}}\, \frac{1}{2\pi}\sqrt{\frac{\mu}{I}}\, \frac{1}{2\pi}\sqrt{\frac{1}{LC}}\, \frac{1}{2\pi}\sqrt{\frac{1}{LC}}\,
Differential equation:
M\ddot x + \gamma\dot x + Kx = F\, I\ddot \theta + \Gamma\dot \theta + \mu \theta = \tau\, L\ddot q + R\dot q + q/C = e\, C\ddot \phi + G\dot \phi + \phi/L = i\,

Application to a conservative force

The problem of the simple harmonic oscillator occurs frequently in physics, because a mass at equilibrium under the influence of any conservative force, in the limit of small motions, behaves as a simple harmonic oscillator.
A conservative force is one that has a potential energy function. The potential energy function of a harmonic oscillator is:
V(x) = \frac{1}{2} k x^2
Given an arbitrary potential energy function V(x), one can do a Taylor expansion in terms of x around an energy minimum (x = x_0) to model the behavior of small perturbations from equilibrium.
V(x) = V(x_0) + (x-x_0) V'(x_0) + \frac{1}{2} (x-x_0)^2 V^{(2)}(x_0) + O(x-x_0)^3
Because V(x_0) is a minimum, the first derivative evaluated at x_0 must be zero, so the linear term drops out:
V(x) = V(x_0) + \frac{1}{2} (x-x_0)^2 V^{(2)}(x_0) + O(x-x_0)^3
The constant term V(x0) is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:
V(x) \approx \frac{1}{2} x^2 V^{(2)}(0) = \frac{1}{2} k x^2
Thus, given an arbitrary potential energy function V(x) with a non-vanishing second derivative, one can use the solution to the simple harmonic oscillator to provide an approximate solution for small perturbations around the equilibrium point.

Examples

Simple pendulum


A simple pendulum exhibits simple harmonic motion under the conditions of no damping and small amplitude.

Assuming no damping and small amplitudes, the differential equation governing a simple pendulum is
{\mathrm{d}^2\theta\over \mathrm{d}t^2}+{g\over \ell}\theta=0.
The solution to this equation is given by:
\theta(t) = \theta_0\cos\left(\sqrt{g\over \ell}t\right) \quad\quad\quad\quad |\theta_0| \ll 1
where \theta_0 is the largest angle attained by the pendulum. The period, the time for one complete oscillation, is given by 2\pi divided by whatever is multiplying the time in the argument of the cosine (\sqrt{g\over \ell} here).
T_0 = 2\pi\sqrt{\ell\over g}\quad\quad\quad\quad |\theta_0| \ll 1.

Pendulum swinging over turntable

Simple harmonic motion can in some cases be considered to be the one-dimensional projection of two-dimensional circular motion. Consider a long pendulum swinging over the turntable of a record player. On the edge of the turntable there is an object. If the object is viewed from the same level as the turntable, a projection of the motion of the object seems to be moving backwards and forwards on a straight line orthogonal to the view direction, sinusoidally like the pendulum.

Spring/mass system


Spring–mass system in equilibrium (A), compressed (B) and stretched (C) states.

When a spring is stretched or compressed by a mass, the spring develops a restoring force. Hooke's law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length:
F \left( t \right) =-kx \left( t \right)
where F is the force, k is the spring constant, and x is the displacement of the mass with respect to the equilibrium position. The minus sign in the equation indicates that the force exerted by the spring always acts in a direction that is opposite to the displacement (i.e. the force always acts towards the zero position), and so prevents the mass from flying off to infinity.

By using either force balance or an energy method, it can be readily shown that the motion of this system is given by the following differential equation:
F(t) = -kx(t) = m \frac {\mathrm{d}^{2}}{\mathrm{d}{t}^{2}} x \left( t \right) = ma.
...the latter being Newton's second law of motion.

If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by:
x \left( t \right) =A \cos \left( \sqrt{k \over m}t \right).
Given an ideal massless spring, m is the mass on the end of the spring. If the spring itself has mass, its effective mass must be included in m.

Energy variation in the spring–damping system

In terms of energy, all systems have two types of energy, potential energy and kinetic energy. When a spring is stretched or compressed, it stores elastic potential energy, which then is transferred into kinetic energy. The potential energy within a spring is determined by the equation U = k{x}^{2}/ 2 .

When the spring is stretched or compressed, kinetic energy of the mass gets converted into potential energy of the spring. By conservation of energy, assuming the datum is defined at the equilibrium position, when the spring reaches its maximum potential energy, the kinetic energy of the mass is zero. When the spring is released, it tries to return to equilibrium, and all its potential energy converts to kinetic energy of the mass.
at

Harmonic


From Wikipedia, the free encyclopedia


The nodes of a vibrating string are harmonics.

Two different notations of natural harmonics on the cello. First as sounded (more common), then as fingered (easier to sightread).

The term harmonic in its strictest sense is any member of the harmonic series. The term is employed in various disciplines, including music and acoustics, electronic power transmission, radio technology, etc. It is typically applied when considering the frequencies of repeating signals, such as sinusoidal waves, that happen to relate as whole-numbered multiples. In that case, a harmonic is a signal whose frequency is a whole-numbered multiple of the frequency of some other given signal. For example, in alternating current of 60 cycles per second, "fifth-harmonic distortion" would produce an unwanted additional signal at 300 cycles per second, exactly five times the original frequency.

A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental frequency, therefore the sum of harmonics is also periodic at that frequency. As multiples of the fundamental frequency, successive harmonics can be found by repeatedly adding the fundamental frequency. For example, if the fundamental frequency (first harmonic) is 25 Hz, the frequencies of the next harmonics are: 50 Hz (2nd harmonic), 75 Hz (3rd harmonic), 100 Hz (4th harmonic) etc.

Characteristics

Most acoustic instruments emit complex tones containing many individual partials (component simple tones or sinusoidal waves), but the untrained human ear typically does not perceive those partials as separate phenomena. Rather, a musical note is perceived as one sound, the quality or timbre of that sound being a result of the relative strengths of the individual partials.

Many acoustic oscillators, such as the human voice or a bowed violin string, produce complex tones that are more or less periodic, and thus are composed of partials that are near matches to integer multiples of the fundamental frequency and therefore resemble the ideal harmonics and are called "harmonic partials" or simply "harmonics" for convenience (although it's not strictly accurate to call a partial a harmonic, the first being real and the second being ideal). Oscillators that produce harmonic partials behave somewhat like 1-dimensional resonators, and are often long and thin, such as a guitar string or a column of air open at both ends (as with the modern orchestral transverse flute). Wind instruments whose air column is open at only one end, such as trumpets and clarinets, also produce partials resembling harmonics. However they only produce partials matching the odd harmonics, at least in theory. The reality of acoustic instruments is such that none of them behaves as perfectly as the somewhat simplified theoretical models would predict.

Partials whose frequencies are not integer multiples of the fundamental are referred to as inharmonic partials.

Some acoustic instruments emit a mix of harmonic and inharmonic partials but still produce an effect on the ear of having a definite fundamental pitch, such as pianos, strings plucked pizzicato, vibraphones, marimbas, and certain pure-sounding bells or chimes. Antique singing bowls are known for producing multiple harmonic partials or multiphonics. [1] [2]

Other oscillators, such as cymbals, drum heads, and other percussion instruments, naturally produce an abundance of inharmonic partials and do not imply any particular pitch, and therefore cannot be used melodically or harmonically in the same way other instruments can.

Harmonics and overtones

An overtone is any frequency higher than the fundamental. The tight relation between overtones and harmonics in music often leads to their being used synonymously in a strictly musical context, but they are counted differently leading to some possible confusion. This chart demonstrates how they are counted:

Frequency Order Name 1 Name 2 Wave Representation Molecular Representation
1 · f =   440 Hz n = 1 fundamental tone 1st harmonic Pipe001.gif Molecule1.gif
2 · f =   880 Hz n = 2 1st overtone 2nd harmonic Pipe002.gif Molecule2.gif
3 · f = 1320 Hz n = 3 2nd overtone 3rd harmonic Pipe003.gif Molecule3.gif
4 · f = 1760 Hz n = 4 3rd overtone 4th harmonic Pipe004.gif Molecule4.gif
In many musical instruments, it is possible to play the upper harmonics without the fundamental note being present.
In a simple case (e.g., recorder) this has the effect of making the note go up in pitch by an octave, but in more complex cases many other pitch variations are obtained. In some cases it also changes the timbre of the note. This is part of the normal method of obtaining higher notes in wind instruments, where it is called overblowing. The extended technique of playing multiphonics also produces harmonics. On string instruments it is possible to produce very pure sounding notes, called harmonics or flageolets by string players, which have an eerie quality, as well as being high in pitch. Harmonics may be used to check at a unison the tuning of strings that are not tuned to the unison. For example, lightly fingering the node found halfway down the highest string of a cello produces the same pitch as lightly fingering the node 1/3 of the way down the second highest string. For the human voice see Overtone singing, which uses harmonics.

While it is true that electronically produced periodic tones (e.g. square waves or other non-sinusoidal waves) have "harmonics" that are whole number multiples of the fundamental frequency, practical instruments do not all have this characteristic. For example higher "harmonics"' of piano notes are not true harmonics but are "overtones" and can be very sharp, i.e. a higher frequency than given by a pure harmonic series. This is especially true of instruments other than stringed or brass/woodwind ones, e.g., xylophone, drums, bells etc., where not all the overtones have a simple whole number ratio with the fundamental frequency.

The fundamental frequency is the reciprocal of the period of the periodic phenomenon.


 
Harmonics on stringed instruments

Playing a harmonic on a string

The following table displays the stop points on a stringed instrument, such as the guitar (guitar harmonics), at which gentle touching of a string will force it into a harmonic mode when vibrated. String harmonics (flageolet tones) are described as having a "flutelike, silvery quality that can be highly effective as a special color" when used and heard in orchestration.[3] It is unusual to encounter natural harmonics higher than the fifth partial on any stringed instrument except the double bass, on account of its much longer strings.[4]
Harmonic Stop note Sounded note relative to open string Cents above open string Cents reduced to one octave Audio
2 octave octave (P8) 1,200.0 0.0 About this sound Play (help·info)
3 just perfect fifth P8 + just perfect fifth (P5) 1,902.0 702.0 About this sound Play (help·info)
4 second octave 2P8 2,400.0 0.0 About this sound Play (help·info)
5 just major third 2P8 + just major third (M3) 2,786.3 386.3 About this sound Play (help·info)
6 just minor third 2P8 + P5 3,102.0 702.0
7 septimal minor third 2P8 + septimal minor seventh (m7) 3,368.8 968.8 About this sound Play (help·info)
8 septimal major second 3P8 3,600.0 0.0
9 Pythagorean major second 3P8 + Pythagorean major second (M2) 3,803.9 203.9 About this sound Play (help·info)
10 just minor whole tone 3P8 + just M3 3,986.3 386.3
11 greater undecimal neutral second 3P8 + lesser undecimal tritone 4,151.3 551.3 About this sound Play (help·info)
12 lesser undecimal neutral second 3P8 + P5 4,302.0 702.0
13 tridecimal 2/3-tone 3P8 + tridecimal neutral sixth (n6) 4,440.5 840.5 About this sound Play (help·info)
14 2/3-tone 3P8 + P5 + septimal minor third (m3) 4,568.8 968.8
15 septimal (or major) diatonic semitone 3P8 + just major seventh (M7) 4,688.3 1,088.3 About this sound Play (help·info)
16 just (or minor) diatonic semitone 4P8 4,800.0 0.0

Table


Table of harmonics of a stringed instrument with colored dots indicating which positions can be lightly fingered to generate just intervals up to the 7th harmonic

Artificial harmonics

Although harmonics are most often used on open strings, occasionally a score will call for an artificial harmonic, produced by playing an overtone on a stopped string. As a performance technique, it is accomplished by using two fingers on the fingerboard, the first to shorten the string to the desired fundamental, with the second touching the node corresponding to the appropriate harmonic.

Other information

Harmonics may be either used or considered as the basis of just intonation systems.

Composer Arnold Dreyblatt is able to bring out different harmonics on the single string of his modified double bass by slightly altering his unique bowing technique halfway between hitting and bowing the strings.

Composer Lawrence Ball uses harmonics to generate music electronically.
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