**Anonymous**

**·**2016-05-04 15:50:59

Please be patient and read the solution

Ans-2

I assume that you know the basics of beat.

Beats is interference of two or more waves with a little difference in frequency. The rest is same i.e., the medium, the initial phase and the direction of propagation

Consider a beat formed by two waves-

y_{1 }= A sin (wt - kx) and y_{2} = A sin (Wt - kx) [with frequencies w and W]

Equation of beats is the trajectory of the particle at origin.

so, y(net, x=0) = A sinwt + AsinWt= 2A cos[(w-W)t/2] sin[(w+w)t/2]

At t=0, y_{1} = y_{2} = 0 (due to phase)

So, at t=0, y_{net,x=0} = 0, due to phase.

As y(net, x=0) = 2A cos[(w-W)t/2] sin[(w+w)t/2]

and at t=0, sin[(w+w)t/2] is zero, the sine function is phase.

So, particle at origin oscillates with Amplitude = 2A cos[(w-W)t/2 Phase function = sin[(w+w)t/2]

If you observe the plot of beats, the Amplitude (Cosine function) in its period, takes Maximum values when Cos gives +1 or -1. So, in one cycle of Cosine function, there are two Amplitude extremums- Maximum magnitude i.e., Amplitude is maximum twice in one cycle

In a beat, there is only one Amplitude maxima, i.e., Amplitude is maximum only once in once cycle.

If the time period of Amplitude is T(A), time period of beats is T(b)=T(A)/2 as in one cycle of Amplitude function, there are two cycles of beats.

If frequency of Amplitude function is V(A), frequency of beats is V(b) = 2V(A)

Now the question-

waves are

y_{1 }= Asin (wt - kx); w=2Î v

y_{2} = Asin (w_{2}t - kx); w_{2}=2Î (v-1) = 2Î v - 2Î = w - 2Î

y_{3} = Asin (w_{3}t - kx); w_{3 }=2Î (v+1) = 2Î v + 2Î = w + 2Î

As per the derivation;

y(net, x=0) = A [sin wt + sin w_{2}t + sin w_{3}t ]

= A [ sin wt + sin (w - 2Î )t + sin (w + 2Î )t ]

=A [sinwt +2 sin(wt) cos(2Î t)]

=A[1 + cos2Î t] sin wt

Amplitude Function = A[1 + cos2Î t] = f(t)

As the function is simple a cosine function, the period can be found very easily. Let it be T,

f(t)= f(t+T)

or A[1 + cos2Î t] = A[1 + cos {2Î t + 2Î T}

or 2Î T=2Î

or T=1s

the amplitude has a time period of 1 second. As the amplitude is simple cosine formula, the time period of beats T(b) is T/2=1/2s as discussed above

Frequency of beats= 1/T(b)

=2 per second