(a) In class we saw that the ideal seismic source is a "spike." The shape of this theoretical signal is the result of adding many cosines (0-phase) together (Fourier theory). Demonstrate that this works.
Start by using a Excel spreadsheet (or your favorite programming language) to generate two columns of numbers. The first column will consist of a series of values representing time, e.g. -0.1 to 0.1 s with increments of 0.001 seconds The second column will represent the Y-axis and will be:
cosine [(2 pi/ T ) t] function with frequency = 1Hz

Then consider the case of a cosine function with a frequency = 10 Hz, 15 Hz etc. to 250 Hz
As you proceed you will be generating columns with different
Y = cosine (X) results where X = [(2 pi/ T ) t]

Now as you create additional columns representing the y values of the different cosine curves, add the cosine curves together by adding the values across the same row and plot this results. Remember that you are actually adding the cosine waves by doing this. You should be able to approach a symmetrical spike if you include a sufficient range and number of frequencies e.g. 1 to 250 every 25 Hz.

(b) Now, graduate students, I want you to take the above exercise one more step to demonstrate that a pi/2 phase shift will change the signal into a couplet (two-pulse wavelet). This is a little closer to real seismic sources. In order to proceed in this direction start off with the workbook you just used to generate the spike. Then, apply a phase shift of pi/2 radians to each of the column formulae. Then add the results. Do you get what you expected?
Now, lets see if you apply a constant time shift of say 0.05 seconds. What is the result?
Congratulations! You are now able to numerically shape seismic wavelets.

(Due 10/20/2003 at 2.00 p.m.)