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:
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?
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?
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!