Adding Simple Harmonic Waves II

This applet which demonstrates the superposition of two traveling simple harmonic waves. The applet can be used to illustrate phenomena in both dispersive and non-dispersive media. For non-dispersive media, the user must enforce | ω₁/k₁| = |ω₂/k₂| when specifying input values. You can change the amplitudes: A, wave numbers: k = 2*π/λ, and angular frequencies: w = 2*π*f of two waves which will be combined to form a third wave. Recall that v = λ*f, so after you adjust an individual wave it will have a speed of v = ω/k. Positive velocities correspond to motion to the right. To reverse the direction of motion, enter a negative frequency or wave number.

A short theoretical background on this phenomena is availible in dvi, postscript or pdf formats.

Standing Waves

Create Standing waves by creating waves of the same wavelength and frequency traveling in the opposite direction.
Try for example: ω₁ = 0.2, ω₂ = -0.2, k₁ = 0.05, k₂ = 0.05,

Investigate Group and Phase Velocities

The "phase velocity", the velocity of the carrier wave is denoted by v_ph.
v_ph = (ω₁ + ω₂)/(k₁ + k₂).

The "group velocity", the velocity of the the modulation, propagates at the v_g.
v_g = (ω₁ - ω₂)/(k₁ - k₂). which, in general, may be greater than, equal to, or less than the phase velocity of the carrier.

Recall that wave speed is related to the angular frequency and wave number by:
v = ω/k

A nondispersive media is one in which the velocity of a wave is independent of the wavelength
v(k) = const. For example the speed of light in vacuum is a constant c = 3 x 10⁸ m/s²

Note that for nondispersive media, v_g = v_ph = v