Beat and Beat frequency

When two sound of nearly equal frequency are sounded together then periodic rise and fall in sound intensity is heared. This phenomeno is called beat. For this phenomenon the difference between frequencies of two sound wave must not exceed 10hz

Beat frewuency
Let us consider two sound waves with frequency f1 and f2 such that f1-f2 <10 hz are sounded together then at the starting points can be given as
y1 = asinw1t
y2 = asinw2t
By the principal of superpostion the resultant wave will be,
y = y1+y2
y = asinw1t + asinw2t
y = A sin(pi)(f1+f2) * t
This is a periodic function with amplitude
A = 2acos(pi)(f1-f2) *t

CASE I Loud sound
Loud sound will be heard when value of A becomes maximum i.e when
cos(pi)(f1-f2)*t = 1
cos(pi)(f1-f2)*t = cpsn(pi)
f = n/(f1-f2)

loud sound will be heared at f = 0, 1/(f1-f2),2/(f1-f2)
Beat frequency = (f1-f2) hz

CASEII Soft Sound
Soft sound will be heared when amplitude is minimum i.e when
cos(pi)(f1-f2)*t = cos((2n+1)/2)*pi
f = 1/(f1-f2)
Beat frequency = (f1-f2)Hz