- #1
harmonic_lens
- 9
- 0
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
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: