3. Wave Propagation
The velocity of a wave traveling in a string depends inversely on the root of the string's linear mass density in much the same way as the velocity of light traveling in a dielectric depends inversely on the root of the dielectric's relative permittivity (the root of the relative permittivity is the refractive index). A wave passing a reference plane on a string can be described by its displacement and slope in the same way that a light ray is described. A harmonic wavetrain on a string is like a ray in that its displacement is continuous, but it differs in that there is no change in slope at an interface where there is a change in velocity. A string may have a kink, but not one that remains at an interface - Newton's law requires it to move along the string, and the wavetrain is no longer harmonic. At a boundary where the velocity changes, a harmonic wavetrain changes its amplitude and wavelength, but not its displacement, slope, or frequency.
Fig.2 A harmonic wavetrain in a string with reference planes d
apart in a region of constant linear mass density.
Because of the continuity in y and y' across an interface, only one matrix is needed to describe wave propagation, and it can be calculated directly:
Written as a matrix equation,
(3)
In the equations, k has its usual meaning, 2p/l , and note that the derivation makes no assumption about the direction of propagation.
The connection between waves in quantum mechanics and waves on a string is direct because the boundary conditions are identical. The connection between electromagnetic waves and waves on a string is less direct. The electric field of a linearly polarized harmonic wave traveling the z direction can be written:
(4)
and
(5)
At optical frequencies it is usually assumed that dielectric properties change at an interface, but magnetic properties do not. Hence continuity of Hx at an interface requires continuity of Bx and also Ey'. The boundary conditions for an electromagnetic wave therefore take the same form as those for a wave on a string with Ey replacing y. (If the wave is written in terms of the magnetic field, analogous results are obtained, but the magnetic analog of Eq.5 involves D, and n2 appears as a factor in the boundary condition.)