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Group velocity of localized waves

Section 2-2 discusses the superposition of two classical sinusoidal waves of slightly different wave numbers k and k+[delta]k to yield a modulated result whose modulations move with a group velocity. A composition almost as simple as the two-component case is the superposition of three waves y_0, y_1, and y_2 with the same total spread [delta]k in wave number (k = 2[pi]/[lambda]);

(see the attachment for the full equations)

The amplitude A/2 of y_1 and y_2 has been chosen to give the superposition the simplest possible form.
y1+y2 can be written as sin A + sin B= 2sin[(A+B)/2]*cos[(A-B)/2], but for y0 how to add them to y0+(y1+y2)?

A) Express the superposition y_0 + y_1 + y_2 as a single product of trigonometric functions. Sketch the resultant wave for t=0.

B) If [delta]k/k = 10^-2, how many zeros does the waveform have within each region of reinforcement (between adjacent zeros of the envelope)?

C) If the phase velocity [omega]/k - 10 cm/sec, [delta]k/k = 10^-2, and [omega][delta]/[omega] = 10^-3, then what is the group velocity of the waveform?


Solution Preview

Please see the attachment.

We are given the following one-dimensional waves:

(a) We wish to express the superposition as a single product of trigonometric functions and to ...

Solution Summary

We solve some problems involving the superposition and group velocity of sine waves.