Normal mode

In mechanical systems

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 m, attached to three springs, each with spring constant k. They are attached in the following manner, forming a system that is physically symmetric:

where the edge points are fixed and cannot move. Let x₁(t) denote the horizontal displacement of the left mass, and x₂(t) denote the displacement of the right mass.

Denoting acceleration (the second derivative of x(t) with respect to time) as $\ddot{x}$, the equations of motion are:

$${\displaystyle \begin{array}{cc}m{\ddot{x}}_1& =-k{x}_1+k(x_2-x_1)=-2k{x}_1+k{x}_2\\ m{\ddot{x}}_2& =-k{x}_2+k(x_1-x_2)=-2k{x}_2+k{x}_1\end{array}}$$

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

$${\displaystyle \begin{array}{cc}{x}_1(t)& =A_1{e}^{i\omega t}\\ x_2(t)& =A_2{e}^{i\omega t}\end{array}}$$

Substituting these into the equations of motion gives us:

$${\displaystyle \begin{array}{cc}-{\omega }^{2}m{A}_1{e}^{i\omega t}& =-2k{A}_1{e}^{i\omega t}+k{A}_2{e}^{i\omega t}\\ -{\omega }^{2}m{A}_2{e}^{i\omega t}& =k{A}_1{e}^{i\omega t}-2k{A}_2{e}^{i\omega t}\end{array}}$$

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

$${\displaystyle \begin{array}{cc}({\omega }^{2}m-2k)A_1+k{A}_2& =0\\ k{A}_1+({\omega }^{2}m-2k)A_2& =0\end{array}}$$

And in matrix representation:

$${\displaystyle \left[\begin{array}{cc}{\omega }^{2}m-2k& k\\ k& {\omega }^{2}m-2k\end{array}\right]\left(\begin{array}{c}{A}_1\\ A_2\end{array}\right)=0}$$

If the matrix on the left is invertible, the unique solution is the trivial solution (A₁, A₂) = (x₁, x₂) = (0, 0). The non trivial solutions are to be found for those values of ω 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:

$${\displaystyle ({\omega }^{2}m-2k{)}^2-k^2=0}$$

Solving for ω, the two positive solutions are:

$${\displaystyle \begin{array}{cc}{\omega }_1& =\sqrt{\frac{k}{m}}\\ {\omega }_2& =\sqrt{\frac{3k}{m}}\end{array}}$$

Substituting ω₁ into the matrix and solving for (A₁, A₂), yields (1, 1). Substituting ω₂ results in (1, −1). (These vectors are eigenvectors, and the frequencies are eigenvalues.)

The first normal mode is: $${\displaystyle {\overrightarrow{\eta }}_1=\left(\begin{array}{c}{x}_1^1(t)\\ x_2^1(t)\end{array}\right)=c_1\left(\begin{array}{c}1\\ 1\end{array}\right)\cos ({\omega }_{1}t+{\phi }_1)}$$

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

The second normal mode is:

$${\displaystyle {\overrightarrow{\eta }}_2=\left(\begin{array}{c}{x}_1^2(t)\\ x_2^2(t)\end{array}\right)=c_2\left(\begin{array}{c}1\\ -1\end{array}\right)\cos ({\omega }_{2}t+{\phi }_2)}$$

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 c₁, c₂, φ₁, and φ₂ 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. (x, y, z) 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:

$${\displaystyle \mathrm{\Psi }(t)=f(x,y,z)\left[A\cos (\omega t)+B\sin (\omega t)\right]}$$

where 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 (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 f(x, y, z) form of the standing wave. This space-dependence is called a normal mode.

Usually, for problems with continuous dependence on (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 n = 1, 2, 3, ...). 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 ν. This is equivalent to the assumption that all atoms vibrate independently with a frequency ν. Einstein also assumed that the allowed energy states of these oscillations are harmonics, or integral multiples of hν. The spectrum of waveforms can be described mathematically using a Fourier series of sinusoidal density fluctuations (or thermal phonons).

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 compression waves. 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 ν is:

$${\displaystyle E(\nu )=\frac{1}{2}h\nu +\frac{h\nu }{e^{h\nu /k_{\text{B}}T}-1}}$$

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

$${\displaystyle E(\nu )=k_{\text{B}}T\left[1+\frac{1}{12}{\left(\frac{h\nu }{k_{\text{B}}T}\right)}^2+\mathcal{O}\left({\left(\frac{h\nu }{k_{\text{B}}T}\right)}^4\right)\right]}$$

where k_B is the Boltzmann constant.

By knowing the thermodynamic formula,

$${\displaystyle {\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }E}\right)}_{N,V}=\frac{1}{T}}$$

the entropy per normal mode is:

$${\displaystyle \begin{array}{cc}S(\nu )& ={\int }_0^T\frac{d}{dT}E\left(\nu \right)\frac{dT}{T}\\ & =\frac{E\left(\nu \right)}{T}-k_{\text{B}}\log \left(1-e^{-\frac{h\nu }{k_{\text{B}}T}}\right)\end{array}}$$

The free energy is:

$${\displaystyle F(\nu )=E-TS=k_{\text{B}}T\log \left(1-e^{-\frac{h\nu }{k_{\text{B}}T}}\right)}$$

which, for k_BT ≫ hν, tends to:

$${\displaystyle F(\nu )=k_{\text{B}}T\log \left(\frac{h\nu }{k_{\text{B}}T}\right)}$$

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 ν and ν + dν. Allow this number to be f(ν)dν. Since the total number of normal modes is 3N, the function f(ν) is given by:

$${\displaystyle \int f(\nu )d\nu =3N}$$

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

$${\displaystyle U=\int f(\nu )E(\nu )d\nu }$$

In quantum mechanics

Bound states in quantum mechanics are analogous to modes. The waves in quantum systems are oscillations in probability amplitude rather than material displacement. The frequency of oscillation, f, relates to the mode energy by E = hf where h is the Planck constant. Thus a system like an atom consists of a linear combination of modes of definite energy. These energies are characteristic of the particular atom. The (complex) square of the probability amplitude at a point in space gives the probability of measuring an electron at that location. The spatial distribution of this probability is characteristic of the atom.

In seismology

Normal modes are generated in the Earth from long wavelength seismic waves from large earthquakes interfering to form standing waves.

For an elastic, isotropic, homogeneous sphere, spheroidal, toroidal and radial (or breathing) modes arise. Spheroidal modes only involve P and SV waves (like Rayleigh waves) and depend on overtone number n and angular order l but have degeneracy of azimuthal order m. Increasing l concentrates fundamental branch closer to surface and at large l this tends to Rayleigh waves. Toroidal modes only involve SH waves (like Love waves) and do not exist in fluid outer core. Radial modes are just a subset of spheroidal modes with l = 0. The degeneracy does not exist on Earth as it is broken by rotation, ellipticity and 3D heterogeneous velocity and density structure.

It may be assumed that each mode can be isolated, the self-coupling approximation, or that many modes close in frequency resonate, the cross-coupling approximation. Self-coupling will solely change the phase velocity and not the number of waves around a great circle, resulting in a stretching or shrinking of standing wave pattern. Modal cross-coupling occurs due to the rotation of the Earth, from aspherical elastic structure, or due to Earth's ellipticity and leads to a mixing of fundamental spheroidal and toroidal modes.