Suppose one has two audible oscillations having the same amplitude with angular frequencies `omega_1` and `omega_2` :
`y_1(t) = A*sin(omega_1*t)` and `y_2 =A*sin(omega_2*t)`
Now suppose one adds together the oscillations to obtain the resultant:
`y_("tot")(t) =y_1(t)+y_2(t)=A(sin(omega_1*t)+sin(omega_2*t))` (1)
Remember the sum and difference of two angles` ` `a` and `b` :
`sin(a+b) =sin(a)*cos(b) +sin(b)*cos(a)`
`sin(a-b) =sin(a)cos(b) -sin(b)*cos(a)`
If one take `a+b = omega_1*t` and `a-b =omega_2*t` one obtains from (1):
`y_("tot") =2A[sin((omega_1+omega_2)/2*t)*cos((omega_1-omega_2)/2*t)]` (2)
What one person hear is the intensity of the sound and the intensity is a power mediated over a surface
` ``I =P/S =E/(S*t)` .
But we know that the power (energy/time) of an oscillator is proportional to the square of the oscillator amplitude.
`I = C*y_("tot")^2` , where `C` is a constant.
Therefore from (2) a person will hear the following intensity of sound:
Now, if the difference `omega_1-omega_2` is small, one can write the Taylor series expansion corresponding to the function `cos^2(x)` for `x ` small around `x =0` .
(upper terms can be neglected because `x` is small)
Hence the ` `expression (3) will become
Thus, around `t=0` the modulating amplitude of the sound heard will be `(omega_1-omega_2)*t` or in other words equal to the difference of the frequencies of original sounds.