InterviewSolution
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InterviewSolution
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A plane wave propagates along positive x-direction in a homogeneous medium of density `p=200 kg//m^(3)`. Due to propagation of the wave medium particle oscillate. Space density of their oscillation energy is `E=0.16pi^(2) J//m^(3)` and maximum shear strain produced in the mendium is `phi_(0)=8pixx10^(-5)`. if at an instant, phase difference between two particles located at points `(1m,1m,1m)` and `(2m, 2m, 2m,)` is `Deltatheta=144^(@)`, assuming at `t=0` phase of particle at `x=0` to be zero, Equation of wave isA. `y pi 10^(-4)sinpipi2000t-0.8xpi`B. `y pi 10^(-4)sinpipi400t-0.8xpi`C. `y pi 10^(-4)sinpipi100t-8xpi`D. `y pi 10^(-4)sinpipi100t-2xpi` |
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Answer» Correct Answer - b Since, the wave is a plane travelling wave, intensity at every point will be the same, since, initial phase of particle at `x=0` is zero and the wave is travelling along positive `x-` deirection equation of the wave will be of the form `delta=a sin omega (t-(x)/(v))` ...`(i)` Let intensity of the wave be `I`, then space density of oscillation energy of medium particles will be equal to `E=(I)/(v)` But, `I=2pi^(2)n^(2)a^(2)p=0.16pi^(2)J//m^(3)` `a^(2)n^(2)=4xx10^(-4)` or, `an=0.02` shear strain of the medium is `phi=(d)/(dx)delta` Differentiating Eq. (i), `phi=-(aomega)/(v)cosomega(t-(x)/(v))` Modulus of shear strain `f` will be maximum when `cosomega(t-(x)/(v))=+-1` `:.`maximum shear srain `8pixx10^(-5)` `phi_(0)=(aomega)/(v)` but it is equal to `(aomega)/(v)=8pixx10^(-5)` where `omega=2pin` `an=4vxx10^(-5)` ....`(iii)` Solving Eqs. (ii) and (iii), `v=500 m//s` since, the wave is travelling along positive `x-`direction, there, phase difference is give by `Deltatheta=2pi(Deltax)/(lambda)` `Deltax=(x_(2)-x_(1))=(2-1)m=1m` `lambda=(2piDeltax)/(Deltatheta)=2.5 m` But `v=nlambda`,therefore, `n=(v)/(lambda)=200 Hz` substituting `n=200 Hz` in Eq. (ii), `a=1xx10^(-4) m` Angular frequency,`omega=2pi n=400 pi rad//s`. substituting all these values in Eq. (i), `Delta=10^(-4) sinpi(400t-0.8x)m` since, due to propagation of the wave, shear strain is produced in the medium, the wave is a plane transverse wave. |
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