- #1

terryds

- 392

- 13

By addition of transverse wave, I can get a beat.

##

y_1 = A\ \sin (\omega_1 t + kx)\\

y_2 = A\ \sin(\omega_2 t + kx)\\

--------------------------------- + \\

y_1 + y_2 = 2A\ \cos(\frac{\omega_1-\omega_2}{2} \ t) \sin(\frac{\omega_1+\omega_2}{2} \ t + kx)##

So, I get new amplitude as a function of t (A' = 2A cos (((ω_1 - ω_2)/2)t)

##y' = A' \sin(\frac{\omega_1+\omega_2}{2} \ t + kx)##

But, I don't know how to determine the frequency of the beat

If I relate ##y' = A' \sin(\frac{\omega_1+\omega_2}{2} \ t + kx)## to the form y = A sin (ωt + kx), I get

##f=\frac{f_1+f_2}{2}##

But, if I relate the amplitude to the form y = A sin (ωt), I get

##f=\frac{f_1-f_2}{2}##Which one is the beat frequency?

Does those two things have relation to the group velocity and phase velocity? Which one is for group, which one is for phase?

And, I remember the correct formula is ##f_{beat} = |f_1 - f_2|## which I don't know where it come from.

Please help. I really don't get the idea.

##

y_1 = A\ \sin (\omega_1 t + kx)\\

y_2 = A\ \sin(\omega_2 t + kx)\\

--------------------------------- + \\

y_1 + y_2 = 2A\ \cos(\frac{\omega_1-\omega_2}{2} \ t) \sin(\frac{\omega_1+\omega_2}{2} \ t + kx)##

So, I get new amplitude as a function of t (A' = 2A cos (((ω_1 - ω_2)/2)t)

##y' = A' \sin(\frac{\omega_1+\omega_2}{2} \ t + kx)##

But, I don't know how to determine the frequency of the beat

If I relate ##y' = A' \sin(\frac{\omega_1+\omega_2}{2} \ t + kx)## to the form y = A sin (ωt + kx), I get

##f=\frac{f_1+f_2}{2}##

But, if I relate the amplitude to the form y = A sin (ωt), I get

##f=\frac{f_1-f_2}{2}##Which one is the beat frequency?

Does those two things have relation to the group velocity and phase velocity? Which one is for group, which one is for phase?

And, I remember the correct formula is ##f_{beat} = |f_1 - f_2|## which I don't know where it come from.

Please help. I really don't get the idea.

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