Question

find the ratio frequencies of fundamental tone and first overtone in a stretched string

Answer 1

The ratio of frequencies produced by stretched string in case of consecutive harmonic is given by 1 : 2 : 3 : 4 ..... etc
[ Because frequency of nth harmonic is given by v = n/2l√{T/m}, here l is length of string , T is tension in string , m is mass per unit length , so for 1st harmonic ,v = 1/2l√{T/m} , 2nd harmonic , v = 2/2l√{T/m} etc...]

fundamental tone means first harmonic , And first overtone means 2nd harmonic
So, ratio of first and 2nd harmonic = 1 : 2
hence, ratio of frequencies of fundamental tone and first overtone = 1/2 or 1 : 2

Answer 2

Fundamental Frequency or First Harmonic:
The first harmonic or fundamental should at least have an anti node at each end with a node included  between them.Its vibrating Length will be , l= λ/2 λ=2l
Therefore Frequency n1= V/λ=V/2l  -----------(1)
where V is the velocity of sound in air.

Second Harmonic or First Over tone:
The Second Harmonic or first overtone will have at least one more node and anti node than the fundamental frequency.
Vibrating length =l=λThen frequency=n2= V/λ=V/l=2V/2l -----------(2)
By taking ratios of frequencies of first harmonic and second harmonic
 we get :n1:n2=(V/2l)/2V/2ln1:n2=1:2