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Help eliminating parameter for harmonic trig combination

  1. May 28, 2014 #1
    Hey guys, I'm reading the Theory of Sound and I've come to a part in which I'm having trouble double-checking the algebra.

    Suppose we have two harmonic sound waves of equal amplitude traveling directly perpendicular to each other.

    \begin{align} u=acos(2πnt-ε) && v=bcos(2πnt) \end{align}

    They may then combine if t is eliminated to form the general ellipse:

    \begin{equation} \frac{u^2}{a^2}+\frac{v^2}{b^2}-\frac{2uv}{ab}cos(ε)-\sin^2{ε}=0 \end{equation}

    My initial approach was to change forms to:

    \begin{align} \frac{u}{a}=cos(2πnt-ε) && \frac{v}{b}=cos(2πnt) \end{align}

    and then expand the cosine term in the u equation, trying to eventually mold its transcendental functions into forms of \begin{equation} cos(2πnt) \end{equation} so I may then substitute in as \begin{equation} \frac{v}{b} \end{equation}

    After a few hours of expansion and resubstitution, I keep arriving at redundant answers. I tried working backwards from the equation given by changing forms to

    \begin{equation} \frac{u^2}{a^2}+\frac{v^2}{b^2}-\frac{2uv}{ab}cos(ε)-(1-\cos^2{ε})=0 \end{equation}

    and then I tried factoring, but I don't think this is the right approach.

    If anyone has experience with combining transcendental functions and their relations to conics, any advice would be appreciated! Thanks!



    ~HL
     
    Last edited: May 28, 2014
  2. jcsd
  3. May 29, 2014 #2

    LCKurtz

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    Those aren't of equal amplitude unless ##a = b##, if that matters.

    Is that what you are trying to verify?

    You are aware that ##1-\cos^2\epsilon = \sin^2\epsilon##, right? That can't be what is bothering you....?
     
  4. May 30, 2014 #3
    I apologize, I wrote it down wrong. The most general form of the combination assumes a general ellipse for harmonic waves of differing amplitudes. According to the text, when amplitudes are the same, the general ellipse degenerates into a perfect circle. We can assume the periods are equal though. Precisely,

    Yes, I am quite aware of the Pythagorean Identities as well as all the other trig identities. I am just having trouble eliminating t because it is nested within sine and cosine both, and I've never had to combine two first-order trigonometric equations into general ellipse. In my experience in college calculus and algebra (for engineers), professors always seemed to give ellipses and elliptic integrals, etc. a wide berth, labeling them as "too complicated".

    BUT I WANT TO KNOW! hahah :)
     
    Last edited: May 30, 2014
  5. May 30, 2014 #4

    LCKurtz

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    I don't get what your problem is. You have derived the equation you started with and the ##t## is eliminated. What is it that you want that you haven't already done???
     
  6. May 30, 2014 #5
    I want to know the actual steps taken to eliminate the parameter. In other words, if all I had were the two initial equations of u and v, and I wanted to make an elliptic equation without t (not knowing the answer), how would I go about doing it?
     
  7. May 30, 2014 #6

    LCKurtz

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    Take a look at http://scipp.ucsc.edu/~haber/ph5B/addsine.pdf, in the appendix on page 5. He does it for sines but it works just as easily for your problem. Don't let all his subscripts mess you up. Just start with$$
    A \cos(\alpha -\epsilon) + B cos(\alpha)= C\cos(\alpha + \phi)$$to keep it simple and use the real part in his argument.

    [Edit]Nevermind. I was working on two things at once and it isn't immediately obvious the relevance of this to your problem.
     
    Last edited: May 30, 2014
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