In the applet above you will see three plots: f(x,t), g(x,t) and the sum of
these two functions. The functions f(x,t) and g(x,t) may be changed
by typing in a valid formula in the text field and pressing the Change
button. The other buttons control the time evolution of the
functions. Note that the current time is displayed in the upper left
corner and the coordinates are displayed upon left-clicking in one of the plots.
The sum of the two waves shows a beat pattern with regions of constructive and destructive interference. An express for this beat pattern can be obtained by simply adding two waves with the same amplitude. Using the 1/2 angle formulas from trig, one obtains:
where delta k = k1-k2, delta w = w1-w2, kav = k1+k2/2 and wav = w1+w2/2. The resulting wave is a cosine wave of kav wave number and wav angular frequency with an envelope given by the first cosine function in the above equation.
The velocity calculated in Exercise 3 of Section 2a is known as the phase velocity and is the velocity of a point on the wave that has a given phase. The velocity of the envelope is known as the group velocity and is given by vg =delta w/delta k.
© 2000 by Prentice-Hall, Inc. A Pearson Company