Editing Normal mode (section)

== General definitions ==

=== Mode ===
In the [[Wave|wave theory]] of physics and engineering, a '''mode''' in a [[dynamical system]] is a [[standing wave]] state of excitation, in which all the components of the system will be affected sinusoidally at a fixed frequency associated with that mode.

Because no real system can perfectly fit under the standing wave framework, the ''mode'' concept is taken as a general characterization of specific states of oscillation, thus treating the dynamic system in a ''linear'' fashion, in which linear [[superposition principle|superposition]] of states can be performed.

Typical examples include:
* In a mechanical dynamical system, a vibrating rope is the most clear example of a mode, in which the rope is the medium, the stress on the rope is the excitation, and the displacement of the rope with respect to its static state is the modal variable.
* In an acoustic dynamical system, a single sound pitch is a mode, in which the air is the medium, the sound pressure in the air is the excitation, and the displacement of the air molecules is the modal variable.
* In a structural dynamical system, a high tall building oscillating under its most flexural axis is a mode, in which all the material of the building -under the proper numerical simplifications- is the medium, the seismic/wind/environmental solicitations are the excitations and the displacements are the modal variable.
* In an electrical dynamical system, a resonant cavity made of thin metal walls, enclosing a hollow space, for a particle accelerator is a pure standing wave system, and thus an example of a mode, in which the hollow space of the cavity is the medium, the RF source (a Klystron or another RF source) is the excitation and the electromagnetic field is the modal variable.
* When relating to [[music]], normal modes of vibrating instruments (strings, air pipes, drums, etc.) are called "[[overtones]]".

The concept of normal modes also finds application in other dynamical systems, such as [[optics]], [[quantum mechanics]], [[atmospheric dynamics]] and [[molecular dynamics]].

Most dynamical systems can be excited in several modes, possibly simultaneously. Each mode is characterized by one or several frequencies,{{dubious|reason=wouldn't that be a superposition of two modes?|date=April 2020}} according to the modal variable field. For example, a vibrating rope in 2D space is defined by a single-frequency (1D axial displacement), but a vibrating rope in 3D space is defined by two frequencies (2D axial displacement).

For a given amplitude on the modal variable, each mode will store a specific amount of energy because of the sinusoidal excitation.

The ''normal'' or ''dominant'' mode of a system with multiple modes will be the mode storing the minimum amount of energy for a given amplitude of the modal variable, or, equivalently, for a given stored amount of energy, the dominant mode will be the mode imposing the maximum amplitude of the modal variable.

===Mode numbers===
A mode of vibration is characterized by a modal frequency and a mode shape. It is numbered according to the number of half waves in the vibration.  For example, if a vibrating beam with both ends pinned displayed a mode shape of half of a sine wave (one peak on the vibrating beam) it would be vibrating in mode 1. If it had a full sine wave (one peak and one trough) it would be vibrating in mode 2.

In a system with two or more dimensions, such as the pictured disk, each dimension is given a mode number.  Using [[polar coordinate system|polar coordinates]], we have a radial coordinate and an angular coordinate. If one measured from the center outward along the radial coordinate one would encounter a full wave, so the mode number in the radial direction is 2. The other direction is trickier, because only half of the disk is considered due to the anti-symmetric (also called [[Symmetry in mathematics#Skew-symmetry|skew-symmetry]]) nature of a disk's vibration in the angular direction. Thus, measuring 180° along the angular direction you would encounter a half wave, so the mode number in the angular direction is 1. So the mode number of the system is 2–1 or 1–2, depending on which coordinate is considered the "first" and which is considered the "second" coordinate (so it is important to always indicate which mode number matches with each coordinate direction).

In linear systems each mode is entirely independent of all other modes.  In general all modes have different frequencies (with lower modes having lower frequencies) and different mode shapes.

===Nodes===
[[File:Mode Shape of a Round Plate with Node Lines.jpg|right|thumb|220px|A mode shape of a drum membrane, with nodal lines shown in pale green]]
In a one-dimensional system at a given mode the vibration will have nodes, or places where the displacement is always zero.  These nodes correspond to points in the mode shape where the mode shape is zero.  Since the vibration of a system is given by the mode shape multiplied by a time function, the displacement of the node points remain zero at all times.

When expanded to a two dimensional system, these nodes become lines where the displacement is always zero.  If you watch the animation above you will see two circles (one about halfway between the edge and center, and the other on the edge itself) and a straight line bisecting the disk, where the displacement is close to zero.  In an idealized system these lines equal zero exactly, as shown to the right.