The frequency (`nu`) of transverse wave on a string may depend upon (i) length L of string (ii) tension T in the string and (iii) mass per unit length m of the string Derive the formula for frequency with the help of dimensions.

Explanation

Let the frequency of vibration of string depend upon L, m and T in the following way:
`mu prop L^(a)`
`nu prop m^(t)`
`nu prop T^( c)`
Combining them `nu prop L^(a)m^(t) T^( c)`
or, `nu = kL^(a)m^(t)T^( c)`
Putting dimensions on both the sides
`(T^(-1)) = (L)^(a) (ML^(-1))^(t)(MLT^(-2))^(c)`
or `(M^(0)L^(0)T^(-1)) = (M^(t+c)L^(a-t+c)T^(-2c))`
Equating powers of M,L,T we get
t+c = 0 , a-t +c=0, -2c =-1
On solving we get a = -1, `t= (-1)/(2),c = (1)/(2)`
Putting value of a,b,c in equation (4), v =`(k)/(L) sqrt((T)/(m))`