# Exercises for Chaos Under Control

## Chapter 6: the Spectra of Noise

4.[N] Suppose S1(t) is a sine wave of amplitude 1 and frequency 1 Hz, and S2(t) is a sine wave of amplitude 2 and frequency 2/3 Hz. Suppose also that both are zero at t = 0 and start to increase as t begins to increase from 0. (That is, S1(t) = sin(6.28t) and S2(t) = sin(4.19t). How did the 6.28 and 4.19 get here? Remember sin(x) repeats after x increases from 0 to 2p (about 6.28) radians, so S1(t) repeats after t increases from 0 to 1.)

(a) What are the values of S1(t) at t = 0.75, 1.5, 2.25, 3, 3.75, and 4.5 seconds? Answer

(b) What are the values of S2(t) at t = 0.75, 1.5, 2.25, 3, 3.75, and 4.5 seconds? Answer

(c) Denote by S3(t) the sum S1(t) + S2(t). What are the values of S3(t) at t = 0.75, 1.5, 2.25, 3, 3.75, and 4.5 seconds? Answer

(d) Guess the frequency of S3(t). Answer

5. Suppose S1(t) is a sine wave of amplitude 1 and frequency 1 Hz, and S2(t) is a sine wave of amplitude 2 and frequency 1/sqrt(2) Hz. Suppose also that both are zero at t = 0 and start to increase as t begins to increase from 0. Recall sqrt(2) is an irrational number, so we claim in the text S3(t) = S1(t) + S2(t) never repeats. On the other hand, observe that S3(0) = 0 and S3(0.390045...) = 0. What's going on here? Answer