Editing Normal mode (section)

==In mechanical systems==
In the analysis of [[conservative system]]s with small displacements from equilibrium, important in [[acoustics]], [[molecular spectra]], and [[electrical circuit]]s, the system can be transformed to new coordinates called '''normal coordinates.'''  Each normal coordinate corresponds to a single vibrational frequency of the system and the corresponding motion of the system is called the normal mode of vibration.<ref name=Goldstein3>{{Cite book |last1=Goldstein |first1=Herbert |title=Classical mechanics |last2=Poole |first2=Charles P. |last3=Safko |first3=John L. |date=2008 |publisher=Addison Wesley |isbn=978-0-201-65702-9 |edition=3rd ed., [Nachdr.] |location=San Francisco, Munich}}</ref>{{rp|p=332}} 
=== Coupled oscillators ===
Consider two equal bodies (not affected by gravity), each of [[mass]] {{mvar|m}}, attached to three springs, each with [[spring constant]] {{mvar|k}}. They are attached in the following manner, forming a system that is physically symmetric:

[[File:Coupled Harmonic Oscillator.svg|300px|center]]

where the edge points are fixed and cannot move. Let {{math|''x''{{sub|1}}(''t'')}} denote the horizontal [[displacement (distance)|displacement]] of the left mass, and {{math|''x''{{sub|2}}(''t'')}} denote the displacement of the right mass.

Denoting acceleration (the second [[derivative]] of {{math|''x''(''t'')}} with respect to time) as {{nowrap|<math display=inline>\ddot x</math>,}} the [[equations of motion]] are:

<math display="block">\begin{align}
  m \ddot x_1 &= - k x_1 + k (x_2 - x_1) = - 2 k x_1 + k x_2 \\
  m \ddot x_2 &= - k x_2 + k (x_1 - x_2) = - 2 k x_2 + k x_1
\end{align}</math>

Since we expect oscillatory motion of a normal mode (where {{mvar|ω}} is the same for both masses), we try:

<math display="block">\begin{align}
  x_1(t) &= A_1 e^{i \omega t} \\
  x_2(t) &= A_2 e^{i \omega t}
\end{align}</math>

Substituting these into the equations of motion gives us:

<math display="block">\begin{align}
  -\omega^2 m A_1 e^{i \omega t} &= - 2 k A_1 e^{i \omega t} + k A_2 e^{i \omega t} \\
  -\omega^2 m A_2 e^{i \omega t} &= k A_1 e^{i \omega t} - 2 k A_2 e^{i \omega t}
\end{align}</math>

Omitting the exponential factor (because it is common to all terms) and simplifying yields:

<math display="block">\begin{align}
  (\omega^2 m - 2 k) A_1 + k A_2 &= 0 \\
  k A_1 + (\omega^2 m - 2 k) A_2 &= 0
\end{align}</math>

And in [[matrix (mathematics)|matrix]] representation:

<math display="block">\begin{bmatrix}
  \omega^2 m - 2 k & k \\
                 k & \omega^2 m - 2 k
  \end{bmatrix} \begin{pmatrix}
    A_1 \\
    A_2
  \end{pmatrix} = 0
</math>

If the matrix on the left is invertible, the unique solution is the trivial solution {{math|1=(''A''{{sub|1}}, ''A''{{sub|2}}) = (''x''{{sub|1}}, ''x''{{sub|2}}) = (0, 0)}}. The non trivial solutions are to be found for those values of {{mvar|ω}} whereby the matrix on the left is [[singular matrix|singular]]; i.e. is not invertible. It follows that the [[determinant]] of the matrix must be equal to 0, so:

<math display="block"> (\omega^2 m - 2 k)^2 - k^2 = 0 </math>

Solving for {{mvar|ω}}, the two positive solutions are:

<math display="block">\begin{align}
  \omega_1 &= \sqrt{\frac{k}{m}} \\
  \omega_2 &= \sqrt{\frac{3 k}{m}}
\end{align}</math>

Substituting {{math|''ω''{{sub|1}}}} into the matrix and solving for {{math|(''A''{{sub|1}}, ''A''{{sub|2}})}}, yields {{math|(1, 1)}}. Substituting {{math|''ω''{{sub|2}}}} results in {{math|(1, −1)}}. (These vectors are [[eigenvector]]s, and the frequencies are [[eigenvalue]]s.)

The first normal mode is:
<math display="block">\vec \eta_1 = \begin{pmatrix}
    x^1_1(t) \\ x^1_2(t)
  \end{pmatrix} = c_1 \begin{pmatrix}1 \\ 1\end{pmatrix} \cos{(\omega_1 t + \varphi_1)}
</math>

Which corresponds to both masses moving in the same direction at the same time. This mode is called antisymmetric.

The second normal mode is:

<math display="block">\vec \eta_2 = \begin{pmatrix}
    x^2_1(t) \\ x^2_2(t)
  \end{pmatrix} = c_2 \begin{pmatrix} 1 \\ -1 \end{pmatrix} \cos{(\omega_2 t + \varphi_2)}
</math>

This corresponds to the masses moving in the opposite directions, while the center of mass remains stationary. This mode is called symmetric.

The general solution is a [[Superposition principle|superposition]] of the '''normal modes''' where {{math|''c''{{sub|1}}}}, {{math|''c''{{sub|2}}}}, {{math|''φ''{{sub|1}}}}, and {{math|''φ''{{sub|2}}}} are determined by the [[initial condition]]s of the problem.

The process demonstrated here can be generalized and formulated using the formalism of [[Lagrangian mechanics]] or [[Hamiltonian mechanics]].

=== Standing waves ===
A [[standing wave]] is a continuous form of normal mode. In a standing wave, all the space elements (i.e. {{math|(''x'', ''y'', ''z'')}} coordinates) are oscillating in the same [[frequency]] and in [[phase (waves)|phase]] (reaching the [[mechanical equilibrium|equilibrium]] point together), but each has a different amplitude.

[[File:Standing-wave05.png]]

The general form of a standing wave is:

<math display="block">
\Psi(t) = f(x,y,z) (A\cos(\omega t) + B\sin(\omega t))
</math>

where {{math|''f''(''x'', ''y'', ''z'')}} represents the dependence of amplitude on location and the cosine/sine are the oscillations in time.

Physically, standing waves are formed by the [[Interference (wave propagation)|interference]] (superposition) of waves and their reflections (although one may also say the opposite; that a moving wave is a [[superposition principle|superposition]] of standing waves). The geometric shape of the medium determines what would be the interference pattern, thus determines the {{math|''f''(''x'', ''y'', ''z'')}} form of the standing wave. This space-dependence is called a '''normal mode'''.

Usually, for problems with continuous dependence on {{math|(''x'', ''y'', ''z'')}} there is no single or finite number of normal modes, but there are infinitely many normal modes. If the problem is bounded (i.e. it is defined on a finite section of space) there are [[countably many]] normal modes (usually numbered {{math|1=''n'' = 1, 2, 3, ...}}). If the problem is not bounded, there is a continuous spectrum of normal modes.

=== Elastic solids ===
{{Main|Einstein solid|Debye model}}

In any solid at any temperature, the primary particles (e.g. atoms or molecules) are not stationary, but rather vibrate about mean positions. In insulators the capacity of the solid to store thermal energy is due almost entirely to these vibrations. Many physical properties of the solid (e.g. modulus of elasticity) can be predicted given knowledge of the frequencies with which the particles vibrate. The simplest assumption (by Einstein) is that all the particles oscillate about their mean positions with the same natural frequency {{mvar|ν}}. This is equivalent to the assumption that all atoms vibrate independently with a frequency {{mvar|ν}}. Einstein also assumed that the allowed energy states of these oscillations are harmonics, or integral multiples of {{mvar|hν}}. The spectrum of waveforms can be described mathematically using a Fourier series of sinusoidal density fluctuations (or thermal [[phonons]]).

[[File:Harmonic partials on strings.svg|thumb|250px|The [[Fundamental frequency|fundamental]] and the first six [[overtone]]s of a vibrating string. The mathematics of [[wave propagation]] in crystalline solids consists of treating the [[harmonics]] as an ideal [[Fourier series]] of [[Sine wave|sinusoidal]] density fluctuations (or atomic displacement waves).]]

Debye subsequently recognized that each oscillator is intimately coupled to its neighboring oscillators at all times. Thus, by replacing Einstein's identical uncoupled oscillators with the same number of coupled oscillators, Debye correlated the elastic vibrations of a one-dimensional solid with the number of mathematically special modes of vibration of a stretched string (see figure). The pure tone of lowest pitch or frequency is referred to as the fundamental and the multiples of that frequency are called its harmonic overtones. He assigned to one of the oscillators the frequency of the fundamental vibration of the whole block of solid. He assigned to the remaining oscillators the frequencies of the harmonics of that fundamental, with the highest of all these frequencies being limited by the motion of the smallest primary unit.

The normal modes of vibration of a crystal are in general superpositions of many overtones, each with an appropriate amplitude and phase. Longer wavelength (low frequency) [[phonons]] are exactly those acoustical vibrations which are considered in the theory of sound. Both longitudinal and transverse waves can be propagated through a solid, while, in general, only longitudinal waves are supported by fluids.

In the [[longitudinal mode]], the displacement of particles from their positions of equilibrium coincides with the propagation direction of the wave. Mechanical longitudinal waves have been also referred to as ''{{dfn|compression waves}}''. For [[transverse mode]]s, individual particles move perpendicular to the propagation of the wave.

According to quantum theory, the mean energy of a normal vibrational mode of a crystalline solid with characteristic frequency {{mvar|ν}} is:

<math display="block">E(\nu) = \frac{1}{2}h\nu + \frac{h\nu}{e^{h\nu/kT} - 1}</math>

The term {{math|(1/2)''hν''}} represents the "zero-point energy", or the energy which an oscillator will have at absolute zero. {{math|''E''(''ν'')}} tends to the classic value {{mvar|kT}} at high temperatures

<math display="block">E(\nu) = kT\left[1 + \frac{1}{12}\left(\frac{h\nu}{kT}\right)^2 + O\left(\frac{h\nu}{kT}\right)^4 + \cdots\right]</math>

By knowing the thermodynamic formula,

<math display="block">\left( \frac{\partial S}{\partial E}\right)_{N,V} = \frac{1}{T}</math>

the entropy per normal mode is:

<math display="block">\begin{align}
  S\left(\nu\right) &= \int_0^T\frac{d}{dT}E\left(\nu\right)\frac{dT}{T} \\[10pt]
      &= \frac{E\left(\nu\right)}{T} - k\log\left(1 - e^{-\frac{h\nu}{kT}}\right)
\end{align}</math>

The free energy is:

<math display="block">F(\nu) = E - TS=kT\log \left(1-e^{-\frac{h\nu}{kT}}\right)</math>

which, for {{math|''kT'' ≫ ''hν''}}, tends to:

<math display="block">F(\nu) = kT\log \left(\frac{h\nu}{kT}\right)</math>

In order to calculate the internal energy and the specific heat, we must know the number of normal vibrational modes a frequency between the values {{mvar|ν}} and {{math|''ν'' + ''dν''}}. Allow this number to be {{math|''f''(''ν'')''dν''}}. Since the total number of normal modes is {{math|3''N''}}, the function {{math|''f''(''ν'')}} is given by:

<math display="block">\int f(\nu)\,d\nu = 3N</math>

The integration is performed over all frequencies of the crystal. Then the internal energy {{mvar|U}} will be given by:

<math display="block">U = \int f(\nu)E(\nu)\,d\nu</math>