String- 1 is connected with string- 2 . The mass per unit length in string- 1 is ( mu_{1} ) and the mass per unit length in string-2 is ( 4 mu_{1} ). The tension in the strings is T. A travelling wave is coming from the left. What fraction of the energy per sec in the incident wave goes into string-2 ? 1) ( 1 / 8 ) 2) ( 4 / 9 ) 3) ( 2 / 3 ) 4) ( 8 / 9 )
Question
String- 1 is connected with string- 2 . The mass per unit length in string- 1 is ( mu_{1} ) and the mass per unit length in string-2 is ( 4 mu_{1} ). The tension in the strings is T. A travelling wave is coming from the left. What fraction of the energy per sec in the incident wave goes into string-2 ? 1) ( 1 / 8 ) 2) ( 4 / 9 ) 3) ( 2 / 3 ) 4) ( 8 / 9 )
Solution
The energy transmitted from one medium to another depends on the impedance of the two media. The impedance (Z) of a medium is given by the square root of the tension (T) divided by the mass per unit length (mu).
For string 1, the impedance Z1 is sqrt(T/mu1). For string 2, the impedance Z2 is sqrt(T/4mu1) = sqrt(T/mu1)/2 = Z1/2.
The fraction of the energy transmitted from one medium to another is given by the formula 4Z1Z2/(Z1+Z2)^2.
Substituting Z1 and Z2 into this formula gives 4Z1(Z1/2)/((Z1+Z1/2)^2) = 2Z1^2/(2.5Z1)^2 = 2/2.5^2 = 2/6.25 = 8/25.
Therefore, the fraction of the energy per second in the incident wave that goes into
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