If we listen, a few minutes apart, two sounds of very close frequencies, say 256 Hz and 260 Hz, we will not be able to discriminate between them. However, if both these sounds reach our ears simultaneously, what we hear is a sound of frequency 258 Hz, the average of the two combining frequencies. In addition we hear a striking variation in the intensity of sound it increases and decreases in slow, wavering beats that repeat at a frequency of 4 Hz, the difference between the frequencies of two incoming sounds. The phenomenon of wavering of sound intensity when two waves of nearly same frequencies and amplitudes travelling in the same direction, are superimposed on each other is called beats.

This phenomenon is observed on adding waves that differ in frequency only.
Consider two waves of same amplitude and phase but with slightly different frequencies `f_1` and `f_2` as described by the the equation

`y_1= a sin(k_1x-omega_1t)`

`y_2= a sin(k_2x-omega_2t)`

Note that different frequencies corresponds to different wave number `k_1 =(omega_1)/v` and `k_2= (omega_2)/v`

Using superposition principle
`y = y_1 + y_2 = a [sin (k_1x- omega_1t) +sin (k_2 x - omega_2t)]`
or `y = 2a cos [1/2(k_1 - k_2)x-1/2(omega_1 - omega_2 )t] sin [1/2(k_1 + k_2)x - 1/2(omega_1 + omega_2)t]`...........(1)

In order to intercept equation (1) let us see how this wave look at `t = 0`.
We define the average wave number `k_(avg) = (k_1 + k_2)/2` and the wave number difference `Deltak = k_1- k_2`

At t = 0, the equation reduces to

`y = [ 2acos((Deltakx)/2)] sin (k_(avg) x)`........(2)
Equation (2) represents a sinusoidal wave with a wave number equal to the average of the two component waves. The amplitude of this wave varies with position `x` as given by
`A = 2acos((Deltakx)/2)` At `x = 0` it is a maximum `2a`, at `x_(min) =pi/(Deltak)` it decreases to zero, and at `x_(max)=(2pi)/(Deltak)` it again reaches a maximum.

As `t` increases from zero in equation (1) the whole pattern shown in figure moves along the x-axis with the wave velocity. At any point in space along the wave path the wave amplitude oscillates in time from a maximum of `2a` to a minimum of zero or - 2a. These pulsation of the amplitude are called beats. The number of amplitude pulsations per second is called the beat frequency. The rime between amplitude maximum for a wave travelling with velocity `v` is
`T = x_(max)/v = (2pi//Deltak)/v`

`therefore` beat frequency `f= 1/T =(v Deltak)/(2pi) =(vk_1)/(2pi)-(vk_2)/(2pi)`
or `f=v/lambda_1-v/lambda_2`
`f_(beat)= f_1-f_2`