Question

When two waves of almost equal frequencies f1 and f2 reach at a point simultaneously, what is the time intetval between successive maxima?

Answer 1

Suppose the two waves have w1 and w2 as their angular frequencies.

Suppose we observe the particle at x=0

The superposition will come out to be

y(net) = y1+y2 = A sin w1*t + A sinw2*t
= 2A sin(w1+w2)/2*t * cos(w1-w2)/2*t

Here, w1 + w2/2 would come out to be a very big value, so it's graph would make the influence on the frequency of the beats.
However, on graph observations, the cos w1-w2/2*t term would make influence on the number of beats and it's time interval for maxima.

Now,for maxima:-
cos (w1-w2)/2*t = +-1
Therefore,
w1-w2/2*t = 0,π,2π,3π.....

2π{f1-f2} / 2 * t = nπ

So,
(f1-f2)*t = n
T = n/f1-f2
So, the time interval between two successive maxima will be 1/f1-f2

So,
Beat frequency = | f1-f2 | beats per second

Hope this helps you !