PHY Problem Set 4

1. Nonlinearity

Background: In Lecture 18, I show the nonlinear effect using a simple example: the

damped/driven nonlinear pendulum, described as equation (1) in Lecture 18 note.

The nonlinear term there is sin(0). I show in page 4 there that sin(0) can be expanded

by Taylor expansion into 0 - 06+ H.O.T (high order terms). The first term 0 is the

linear term while all other are the nonlinear terms. If we assume sin(0) is dominated

by the linear term 0, we use sin(0) = 0 and solve equation (1), which is the simple

harmonics equation that you are familiar with, and this would cause a solution of 0

that is linearly proportional to cos(wa t), denoted as 0~cos(wat). Now consider the

nonlinear effect: Putting ~ cos(wa t) into the first nonlinear term -03/6 in Taylor

expansion above, page 4 shows that this would cause a term of new frequency 3wd.

Question: Consider the next nonlinear term (the one higher than -0³/6) in H.O.T

above, and still use 0~cos(wa t). Shows what new frequency (or frequencies) will

this term cause. You need to show the math derivation rigorously. Hint: you may

use high school math...