1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proof of sinusoidal periodicity

  1. Oct 23, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that
    [ltex]
    f\left(x\right) = \cos(x) + \cos\left(\alpha x \right)
    [/ltex]
    where alpha is a rational number, is a periodic function.

    EDIT: Also, what is it's period?

    2. Relevant equations
    [ltex]
    f\left(x\right + p) = f\left(x\right)
    [/ltex]
    trig identities

    3. The attempt at a solution
    First, I used the definition of periodicity, then trig identities and term collection to get
    [ltex]
    \cos x \cos p - \sin x \sin p - \cos x = \cos \alpha x - \cos \alpha x \cos \alpha p + \sin \alpha x \sin \alpha p
    [/ltex]
    Since p is a constant (if it exists) and the left side is periodic by definition, the right side and hence the function must be periodic as well, yes?

    EDIT: Now for the period, it would seem to need to be greater than 2 pi due to the naked cosine out in front. So, I would suppose that it would be something like [ltex]2 \pi + \frac{2 \pi}{\alpha}[/ltex], right?

    EDIT2: WTH is going on, the board keeps eating my latex! (much later) and then it starts working again. How bizarre.
     
    Last edited: Oct 23, 2009
  2. jcsd
  3. Oct 23, 2009 #2
    Is alpha an integer, a fraction, or a real number?
     
  4. Oct 23, 2009 #3
    Duh, the one thing I forgot to put in the post. It's a rational number.
     
  5. Oct 23, 2009 #4
    OK, good. I'm not sure where applying the trig identity could lead you, but I don't think it will help.

    By way of a hint, for
    [ltex]f\left(x\right) = \cos(x) + \cos\left(\alpha x \right)[/ltex]
    to be periodic
    [tex]\cos(x)[/ltex]
    and
    [ltex]\cos\left(\alpha x \right)[/ltex]
    will each have an integral number of cycles in some unknown interval.
     
  6. Oct 23, 2009 #5
    The point behind applying the identity was to separate p from the interior of the function and allow me to rearrange the definition so that one side was in terms only of x and the other in terms of alpha x. Since I know that the basic functions are 2 pi periodic...

    Anyways, thinking about it your way, I have
    [ltex]n = \frac{k}{\alpha}[/ltex]
    where n is an integer (the number of cycles the plain cos has gone through) and k is also an integer referring to the number of cycles that cos alpha x has gone through. Since alpha is a rational number, it can be decomposed into the general form
    [ltex]\frac{p}{q}[/ltex]
    where p and q are integers. Since k and n are both integers, k = p since otherwise alpha would not reduce to an integer and hence the function must be periodic. QED. Huh, that was more straightforward than I thought.
     
  7. Oct 23, 2009 #6
    I didn't quite follow all of that but I think you've get the notion.

    We could change the form of the equation and write things out in a way that's a little easier on the eyes.

    [tex]f(y) = cos \left(2 \pi \frac{y}{M} \right) + cos \left( 2\pi \frac{y}{N} \right)[/tex]

    When y = pM and y = qN each will have cycled an integral number of times. In this form the problem would have been easier to solve, I think. I'm using variables that are all integers.

    To put it into the form of the given equation

    [tex]2 \pi \frac{y}{M} = x [/tex]

    and

    [tex]2 \pi \frac{y}{N} = \alpha x \ .[/tex]

    What is alpha in terms of M and N?
     
    Last edited: Oct 23, 2009
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proof of sinusoidal periodicity
  1. Sinusoidal Waveforms (Replies: 2)

  2. Sinusoidal functions (Replies: 3)

Loading...