Equations of Motion - Analogues For Waves and Fields

Analogues For Waves and Fields

Field equations

Equations that describe the spatial dependence and time evolution of fields are called field equations. These include

• the Navier–Stokes equations for the velocity field of a fluid,
• Maxwell's equations for the electromagnetic field,
• the Einstein field equation for gravitation (Newton's law of gravity is a special case for weak gravitational fields and low velocities of particles).

Wave equations

Equations of wave motion are called wave equations. The solutions to a wave equation give the time-evolution and spatial dependence of the amplitude. Boundary conditions determine if the solutions describe traveling waves or standing waves.

From classical equations of motion and field equations; mechanical and electromagnetic wave equations can be derived. The general linear wave equation in 3d is:

where X = X(r, t) is any mechanical or electromagnetic field amplitude, say:

• the transverse or longitudinal displacement of a vibrating rod, wire, cable, membrane etc.,
• the fluctuating pressure of a medium, sound pressure,
• the electric fields E or D, or the magnetic fields B or H,
• the voltage V or current I in an alternating current circuit,

and v is the phase velocity. Non-linear equations model the dependence of phase velocity on amplitude, replacing v by v(X). There are other wave equations for very specific applications, non-linear equations arise in different mathematical forms (see for example the Korteweg–de Vries equation).

In quantum mechanics, the analogue of the equation of motion is the Schrödinger equation:

where is the Hamiltonian operator (rather than a function as above), Ψ is the wavefunction and ħ is the reduced Planck constant. Setting up the Hamiltonian and inserting it into the equation results in a differential equation, the solution is the wavefunction as a function of space and time. There are also relativistic wave equations used in quantum field theory.