Superposition

  f(x,t) =

g(x,t) =

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.

Student Exercises:

  1. Measure the wavelength and frequency of the default cosine waves and their envelope.  Compare these results with those predicted by the formula for f(x,t) + g(x,t) using the results from Exercise 1 of Section 2a.

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.

  1. Calculate the group velocity for f(x,t) + g(x,t).  Calculate the phase velocity for the wave within the envelope.  Click the Forward button and describe the progression of the sum wave.
  2. Change g(x,t) to 2*cos(2.6*pi*x-2.2*pi*t).  Click the Forward button and describe the progression of the sum wave.  Calculate the group velocity for the sum wave.  What is the meaning of the phase velocity for a superposition of waves?
  3. Change g(x,t) to 2*cos(2.6*pi*x-2.4*pi*t).  Calculate the group velocity for the sum wave.
  4. Change g(x,t) to 2*cos(2.6*pi*x-2.0*pi*t).  Calculate the group velocity for the sum wave.

 

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