how to prove that the frequencies produced in the string are in the ratio 1:2:3

Asked by lakshmiks703 | 17th Apr, 2022, 07:23: PM

Expert Answer:

When a stretched string vibrates in fundamenetal mode or first harmonic , wavelength λ1 = 2 l  as shown in figure ,
where l is length of string. If v is the speed of wave then frequency n1 of fundamental mode , n1 = v / λ1 = 1 × ( v / 2l ) .
 
When a stretched string vibrates in second harmonic , wavelength λ2 =  l  as shown in figure ,
Then frequency n2 of second harmonic , n2 = v / λ2 = 2 × ( v / 2l ) .
 
Similarly , frequency n3 of third harmonic , n3 = v / λ3 = 3 × ( v / 2l ) .
 
Frequency n4 of fourth harmonic , n4 = v / λ4 = 4 × ( v / 2l ) .
 
Hence frequencies of vibration in stretched string = n1 : n2 : n3 : n4  = 1 × ( v / 2l ) : 2 × ( v / 2l ) : 3 × ( v / 2l ) : 4 × ( v / 2l ) .
 
n1 : n2 : n3 : n4  = 1  : 2  : : 4  .
 

Answered by Thiyagarajan K | 18th Apr, 2022, 09:06: AM