Understanding Beat Frequencies in Sound Waves

[Sho Kano]

Homework Statement

Homework Equations

$f_beat=f_1-f_2$

The Attempt at a Solution

Why are the two different answers? Is it because the first question is asking for how often it fluctuates, and the other is actually asking for the frequency of the sound? Why is the resultant tone the average of the two?

 

[haruspex]

Is it because the first question is asking for how often it fluctuates, and the other is actually asking for the frequency of the sound?

Yes.

Why is the resultant tone the average of the two?

Consider summing two tones of the same amplitude, A sin(ωt)+A sin(ψt). Do you know a way to write that as a product of trig functions?

 

[Sho Kano]

Yes.

Consider summing two tones of the same amplitude, A sin(ωt)+A sin(ψt). Do you know a way to write that as a product of trig functions?

$2A[sin(\frac{wt+ \varphi t}{2})cos(\frac{wt-\varphi t}{2})]$ There's an average in the sine, but not in the cosine, how does this relate to an average freq?

 

[haruspex]

$2A[sin(\frac{wt+ \varphi t}{2})cos(\frac{wt-\varphi t}{2})]$ There's an average in the sine, but not in the cosine, how does this relate to an average freq?

Assuming ψ and ω are similar in value, that product has one frequency as the average of those and the other factor a much lower frequency. Mathematically that does not make them fundamentally different, but to a human observer it will sound and look like a wave of the average frequency with an amplitude varying at the much lower (beat) frequency.