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A wave traveling from the left encounters a junction between the heavy string in which it moves and a lighter one. The incoming wave divides equally into a transmitted and reflected component. What is the ratio of densities of the string?

Asked by anaghaagarwal15 | 15 Mar, 2022, 17:15: PM

Expert Answer

Amplitudes of reflected waves and transmitted waves are given as

begin mathsize 14px style A subscript r space equals space fraction numerator left parenthesis space v subscript 2 space minus space v subscript 1 right parenthesis over denominator left parenthesis space v subscript 1 space plus space v subscript 2 right parenthesis end fraction A subscript i space end subscript end style

............................(1)

begin mathsize 14px style A subscript t space equals space fraction numerator 2 space v subscript 2 over denominator left parenthesis space v subscript 1 plus v subscript 2 right parenthesis end fraction A subscript i end style

..........................(2)

Where A denotes Amplitude of waves , subscript i denotes incident wave,

subscript r denotes reflected wave and subscript t denotes transmitted wave.

v₁ and v₂ are velocities in string-1 and string-2 respectively.

If the incoming wave is divided equally into reflected and transmitted component means

intensities of reflected and transmitted components are equal.

Since intensity of wave is proportional square of amplitude , we get

A_r² = A_t²

Using eqn.(1) and (2) in above expression , we get after simplification as

begin mathsize 14px style left parenthesis space v subscript 2 space minus space v subscript 1 space right parenthesis to the power of 2 space end exponent space equals space 4 space v subscript 2 end subscript superscript 2 end superscript end style

we get the following quadratic equation in v₁ from above expression

begin mathsize 14px style v subscript 1 superscript 2 space minus space 2 space v subscript 1 v subscript 2 space minus space 3 space v subscript 2 superscript 2 space equals space 0 end style

we get v₂ = 3 v₁ as acceptable solution from above expression

velocity of wave in the string is given as

begin mathsize 14px style v space equals space square root of T over d end root end style

where T is tension in the string and d is linear mass density ( mass per unit length )

hence , v²

begin mathsize 14px style proportional to end style

( 1 / d)

begin mathsize 14px style fraction numerator v subscript 2 superscript 2 over denominator v subscript 1 superscript 2 end fraction space equals space d subscript 1 over d subscript 2 space equals space 9 end style

Answered by Thiyagarajan K | 15 Mar, 2022, 23:07: PM

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