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A plane undamped harmonic wave propagates in a medium. Find the mean space density of the total oscillation energy ( :w: ), if at any point of the mediumthe space density of the total oscillation energy becomes equal to w_(0) one- sixth of an oscillation perio after passing the displacement maximum |
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Answer» Solution :The DISPLACEMENT of oscillations is GIVEN by `xi=a COS ( omegat- kx)` Without loss of generality , we confine overselves to ` x=0`. Then the displacement maxima OCCUR at `omegat=npi`. Concentrate at `omegat=0`. Now the energy density is given by `w= rho a^(2) omega^(2) sin^(2) omegat `at ` x=0` `T//6` time later `(` where `T=(pi)/( omega)` is the time period `)` than `t=0`. `w=rho a^(2) omega^(2) sin^(2) ((pi)/( 3))=(3)/(4) rho a^(2) omega^(2)= w _(0)` Thus `lt w gt . = (1)/(2)rho a^(2) omega^(2)=(2w_(0))/( 3)`. |
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