Eliminating time parameter from perpendicular harmonic sound waves

[harmonic_lens]

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

 

[LCKurtz]

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}

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

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}

Is that what you are trying to verify?

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

You are aware that $1-\cos^2\epsilon = \sin^2\epsilon$, right? That can't be what is bothering you...?

 

[harmonic_lens]

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

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,

Consider two harmonic waves traveling in perpendicular directions whose periods are not only able to be expressed as integer ratios, but the ratios involve two small whole numbers.

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 :)

 

[LCKurtz]

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?

 

[harmonic_lens]

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?

 

[LCKurtz]

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.