Question

Waves: does this formula look correct for the superposition of two waves of the same amplitude? How do I derive it?
My two functions are sinusoids with equations
\(sin(k_1*x-*\omega_1*t)\) and \(sin(k_2*x-*\omega_2*t)\)

Answer 1

http://prntscr.com/6pzsvv or if you prefer
\(2\sin (\frac{k_2+k_1}{2}x-\frac{\omega_2+\omega_1}{2}t)\cos (\frac{k_2-k_1}{2}x-\frac{\omega_2-\omega_1}{2}t)\)

Answer 2

oh think I found it
http://web.clark.edu/ggrey/Physics102/Notes/superposition_of_waves.htm

Answer 3

forgot to say that my v is the same in magnitude for both waves

Answer 4

we can start from here:
\[\begin{gathered}
{\psi _1} = A\exp \left[ {i\left( {{k_1}x - {\omega _1}t} \right)} \right] \hfill \\
{\psi _2} = A\exp \left[ {i\left( {{k_2}x - {\omega _2}t} \right)} \right] \hfill \\
\end{gathered} \]
and then we have to compute this:
\[\left( {{\psi _1} + {\psi _2}} \right)\left( {{\psi _1} + {\psi _2}} \right)*\]
where \[\psi *\]
is the complex conjugate of \[\psi \]

Answer 5

@Michele_Laino +1

i was thinking that too.

Answer 6

but looks quite messy, ie isn't really a shortcut.....?

Answer 7

@Michele_Laino what is the meaning of having the star conjugate symbol outside of the parentheses?

Answer 8

I mean this:
\[\left( {{\psi _1} + {\psi _2}} \right)* = {\psi _1}* + {\psi _2}*\]

Answer 9

okay thank you.

Answer 10

thank you!