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# Exercises

1. Verify Equations (701)-(703). Derive Equations (704) and (705) from Equation (699) and Equations (701)-(703).
2. Suppose that

Demonstrate that

where is the square-root of minus one, and . [Hint: You will need to complete the square of the exponent of , transform the variable of integration, and then make use of the standard result that .] Hence, show from Euler's theorem, , that

3. Demonstrate that

4. Verify Equations (728) and (729).
5. Derive Equations (708) and (709) directly from Equation (707) using the results (727)-(729).
6. Verify directly that Equation (746) is a solution of the wave equation, (730), for arbitrary pulse shapes and .
7. Verify Equation (750).

8. Consider a function that is zero for negative , and takes the value for . Find its Fourier transforms, and , defined in

[Hint: Use Euler's theorem.]

9. Let be zero, except in the interval from to . Suppose that in this interval makes exactly one sinusoidal oscillation at the angular frequency , starting and ending with the value zero. Find the previously defined Fourier transforms and .

10. Demonstrate that

where the relation between , , and is as previously defined. This result is known as Parseval's theorem.

11. Suppose that and are both even functions of with the cosine transforms and , so that

Let , and let be the cosine transform of this even function, so that

Demonstrate that

This result is known as the convolution theorem, since the above type of integral is known as a convolution integral. Suppose that . Show that

12. Demonstrate that

where is an arbitrary function, and a constant unit vector, is a solution of the three-dimensional wave equation, (537). How would you interpret this solution?
13. Demonstrate that

where is an arbitrary function, is a solution of the spherical wave equation, (540). How would you interpret this solution?

Next: Dispersive Waves Up: Wave Pulses Previous: Bandwidth
Richard Fitzpatrick 2013-04-08