In the analysis of
conservative systems with small displacements from equilibrium, important in
acoustics,
molecular spectra, and
electrical circuits, 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.
Coupled oscillators Consider two equal bodies (not affected by gravity), each of
mass , attached to three springs, each with
spring constant . They are attached in the following manner, forming a system that is physically symmetric: where the edge points are fixed and cannot move. Let denote the horizontal
displacement of the left mass, and denote the displacement of the right mass. Denoting acceleration (the second
derivative of with respect to time) as the
equations of motion are: \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} Since we expect oscillatory motion of a normal mode (where is the same for both masses), we try: \begin{align} x_1(t) &= A_1 e^{i \omega t} \\ x_2(t) &= A_2 e^{i \omega t} \end{align} Substituting these into the equations of motion gives us: \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} Omitting the exponential factor (because it is common to all terms) and simplifying yields: \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} And in
matrix representation: \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 If the matrix on the left is invertible, the unique solution is the trivial solution . The non trivial solutions are to be found for those values of whereby the matrix on the left is
singular; i.e. is not invertible. It follows that the
determinant of the matrix must be equal to 0, so: (\omega^2 m - 2 k)^2 - k^2 = 0 Solving for , the two positive solutions are: \begin{align} \omega_1 &= \sqrt{\frac{k}{m}} \\ \omega_2 &= \sqrt{\frac{3 k}{m}} \end{align} Substituting into the matrix and solving for , yields . Substituting results in . (These vectors are
eigenvectors, and the frequencies are
eigenvalues.) The first normal mode is: \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)} Which corresponds to both masses moving in the same direction at the same time. This mode is called antisymmetric. The second normal mode is: \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)} 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 of the
normal modes where , , , and are determined by the
initial conditions 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. coordinates) are oscillating in the same
frequency and in
phase (reaching the
equilibrium point together), but each has a different amplitude. The general form of a standing wave is: \Psi(t) = f(x,y,z) (A\cos(\omega t) + B\sin(\omega t)) where represents the dependence of amplitude on location and the cosine/sine are the oscillations in time. Physically, standing waves are formed by the
interference (superposition) of waves and their reflections (although one may also say the opposite; that a moving wave is a
superposition of standing waves). The geometric shape of the medium determines what would be the interference pattern, thus determines the form of the standing wave. This space-dependence is called a
normal mode. Usually, for problems with continuous dependence on 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 ). If the problem is not bounded, there is a continuous spectrum of normal modes.
Elastic solids 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 . This is equivalent to the assumption that all atoms vibrate independently with a frequency . Einstein also assumed that the allowed energy states of these oscillations are harmonics, or integral multiples of . The spectrum of waveforms can be described mathematically using a Fourier series of sinusoidal density fluctuations (or thermal
phonons). and the first six
overtones of a vibrating string. The mathematics of
wave propagation in crystalline solids consists of treating the
harmonics as an ideal
Fourier series of
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 ''''. For
transverse modes, 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 is: E(\nu) = \frac{1}{2}h\nu + \frac{h\nu}{e^{h\nu/kT} - 1} The term represents the "
zero-point energy", or the energy which an oscillator will have at absolute zero. tends to the classic value at high temperatures 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] By knowing the thermodynamic formula, \left( \frac{\partial S}{\partial E}\right)_{N,V} = \frac{1}{T} the entropy per normal mode is: \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} The free energy is: F(\nu) = E - TS=kT\log \left(1-e^{-\frac{h\nu}{kT}}\right) which, for , tends to: F(\nu) = kT\log \left(\frac{h\nu}{kT}\right) 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 and . Allow this number to be . Since the total number of normal modes is , the function is given by: \int f(\nu)\,d\nu = 3N The integration is performed over all frequencies of the crystal. Then the internal energy will be given by: U = \int f(\nu)E(\nu)\,d\nu == In quantum mechanics ==