Molecular mechanics simulations, for example, use a single classical expression for the energy of a compound, for instance the harmonic oscillator.

The vibrational energy is approximately that of a quantum harmonic oscillator:

There are two main types of electronic oscillator: the linear or harmonic oscillator and the nonlinear or relaxation oscillator.

The harmonic, or linear, oscillator produces a sinusoidal output.

For instance, consider a damped harmonic oscillator such as a pendulum, or a resonant L-C tank circuit.

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:

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.

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

A simple harmonic oscillator is an oscillator that is neither driven nor damped.

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 =, the number of cycles per unit time.

The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position but with shifted phases.

The potential energy stored in a simple harmonic oscillator at position x is

A damped harmonic oscillator, which slows down due to friction

In particular, in the Born – Oppenheimer and harmonic approximations, i. e. when the molecular Hamiltonian corresponding to the electronic ground state can be approximated by a harmonic oscillator in the neighborhood of the equilibrium molecular geometry, the resonant frequencies are determined by the normal modes corresponding to the molecular electronic ground state potential energy surface.

As it turns out, analytic solutions of the Schrödinger equation are only available for a very small number of relatively simple model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen molecular ion, and the hydrogen atom are the most important representatives.

For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.

The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year.

F is the fundamental frequency ; the first overtone is the third harmonic, 3F, and the second overtone is the fifth harmonic, 5F for such a pipe, which is a good model for a panflute.

In order to get these numbers, the nuclear shell model starts from an average potential with a shape something between the square well and the harmonic oscillator.

Nevertheless, the magic numbers of nucleons, as well as other properties, can be arrived at by approximating the model with a three-dimensional harmonic oscillator plus a spin-orbit interaction.

Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics.

In the most basic model of a free piston device, the kinematics will result in simple harmonic motion.

Tenney wrote the seminal Meta (+) Hodos ( one of, if not the, earliest applications of gestalt theory and cognitive science to music ), the later Hierarchical temporal gestalt perception in music: a metric space model with Larry Polansky, John Cage and the Theory of Harmony ( 1983, the fullest exposition of his theories of harmonic space ), and other works.

It treats the vibrations of the atomic lattice ( heat ) as phonons in a box, in contrast to the Einstein model, which treats the solid as many individual, non-interacting quantum harmonic oscillators.

Combining this with the expected energy of a harmonic oscillator at temperature T ( already used by Einstein in his model ) would give an energy of

Historically, it is the sales-weighted harmonic mean fuel economy, expressed in miles per US gallon ( mpg ), of a manufacturer's fleet of current model year passenger cars or light trucks with a gross vehicle weight rating ( GVWR ) of 8, 500 pounds ( 3, 856 kg ) or less, manufactured for sale in the US.

The matrix counterparts of the creation and annihilation operators obtained from the quantum harmonic oscillator model are

Using the common rope model of an undamped harmonic oscillator ( HO ) the impact force F < sub > max </ sub > in the rope is given by:

This simple undamped harmonic oscillator model of a rope, however, cannot explain real ropes.

Friction in the rope leads to energy dissipation and thus to a reduction of the impact force compared to the undamped harmonic oscillator model.

Another somewhat related model is the harmonic explorer.

This has been demonstrated by calculations using a modified harmonic oscillator as a model system, in which an exactly solvable system is approached using the variational method.

The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.

Similar to the view of Leopold Kronecker that " God made the integers ; all else is the work of man ," musicians drawn to the alphorn and other instruments that sound the natural harmonics, such as the natural horn, consider the notes of the natural harmonic series — particularly the 7th and 11th harmonics — to be God's Notes, the remainder of the chromatic scale enabled by keys, valves, slides and other methods of changing the qualities of the simple open pipe being an artifact of mere mortals.

It is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal which has been buried under noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies.

If the speed of the fluttering is close to a harmonic of the control's movement, the resonance could break the control off completely.

His next important breakthrough was in the Opus 33 string quartets ( 1781 ), where the melodic and the harmonic roles segue among the instruments: it is often momentarily unclear what is melody and what is harmony.

Evaluation of the related second derivatives allows the prediction of vibrational frequencies if harmonic motion is estimated.

Therefore x ( t ) = cos t. This is an example of simple harmonic motion.

Both natural harmonics and artificial harmonics, where the thumb stops the note and the octave or other harmonic is activated by lightly touching the string at the relative node point, extend the instrument's range considerably.

Of course, if two stations transmit on the same frequency, it is practically impossible for the receiver to separate them ; so instead of all stations transmitting at the same frequency, each chain was allocated a nominal frequency, 1f, and each station in the chain transmitted at a harmonic of this base frequency, as follows:

The euphonium ( like the baritone ; see below for differences ) is pitched in concert B, meaning that when no valves are in use the instrument will produce partials of the B harmonic series.

A harmonic is a frequency that is a whole number multiple of a lower register, or " fundamental " note of the flute.

In other contexts, it is more common to abbreviate it as f < sub > 1 </ sub >, the first harmonic.

( The second harmonic is then f < sub > 2 </ sub > = 2 ⋅ f < sub > 1 </ sub >, etc.

According to Manolo Sanlúcar E is here the tonic, F has the harmonic function of dominant while Am and G assume the functions of subdominant and mediant respectively.

Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis.

which is Laplace's equation, the solutions to which are called harmonic functions by mathematicians.

Schmidt is generally, if erroneously, regarded as a conservative composer ( such labels rest upon yet-to-be-resolved aesthetic / stylistic arguments ), but the rhythmic subtlety and harmonic complexity of much of his music belie this.

This last summation is the harmonic series, which diverges.

The resulting theory is a central part of harmonic analysis.

The geometric mean is also one of the three classical Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean.

For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between ( see Inequality of arithmetic and geometric means.