# Harmonic Motion as a superposition of TWO equal and opposite real circular motions

Im having some trouble understanding the point the author (A.P. French) of the book (Vibrations and Waves) is trying to make in the pages shown below. To be specific, there are a few things that I find confusing:

1. On page 44 (the first picture) the author states

These combine to give a harmonic oscillation along the x axis, if the lengths of C1 and C2 are equal

Where does this requirement come from? This is only true if we want there to ONLY be an x (Re) Component to the solution. But why should this be the case? In a general case, C1 and C2 are NOT always equal, and we still have SHM?

2. The second point the author makes, directly after:

But C1 and C2, as they appear in Eq. (3-7), do not have to be real. We can satisfy Eq. (3-7) just as well if C1 is rotated through some angle α with respect to the direction defined by t, provided that C2 is rotated through -α with respect to -ωt, again making the vector of equal length...

What is meant by "making the vector of equal length?" Since α and ωt are the same for both C1 and C2 (in Fig 3-2 (b)), then C1 and C2 already had to have been of same length to start off with! What is with this talk of "making" the vector of equal length?

3. On page 45 (second picture)

This less restrictive condition then leads to... In what way is this less restrictive? In that C1 and C2 dont have to be real, but can also be complex? In what way is this less restrictive?

I must admit I am confused by the point the author is driving at here. I know I must be missing something, because I have otherwise found this book to be an excellent teacher in this topic, often providing new insights or new ways of looking at seemingly mundane and routine topics. In this case however, I just dont get it. Perhaps Im too used to the regular approach of taking the projection along the x-Axis of just one rotation vector, and I cant step back and look at it more fundamentally.

Any guidance on what you think is being said would be a great help!

• The "reference circle" formulation takes advantage of the same relationship between SHM and circular functions (i.e. sinusoids), though it does away with the "two counter rotating circles" idea and just posits projection out of one direction. Either way the point of the game is to gets $\sin$s and $\cos$s out without having to solve a differential equation directly. – dmckee Apr 2 '17 at 20:04

What you have not done is stated what the title of this section is - "Solving the harmonic oscillator equation using complex exponential".

So he considers a special pair of cases to illustrate why it is that he can say that the solution is the real part of $A e^{j(\omega t +\alpha)}$.

Look what happens when $t=0$.

In the first case $C_1$ and $C_2$ are both along the real axis and so are real (and equal magnitude) constants.

Now with the phase angle $\alpha$ at $t=0$ you have that $C_1$ and $C_2$ are complex and again of equal magnitude.
It is less restrictive because he has shown that these constants can be complex.

He is then using these ideas to present the function $A\cos(\omega t + \alpha)$ as being an equivalent solution and showing it is equivalent to the solutions with the exponential function in them.

Finally he says that this is the real part of $A e^{j(\omega t +\alpha)}$ which will use later on in the book.

• So you're saying that he isn't saying that C1 and C2 have to be equal in Magnitude in general, but rather only if we want the combination to give harmonic motion along the x-axis. – user999318 Apr 2 '17 at 18:52