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InterviewSolution
| 1. |
Two plane wavespropagate in a homogeneous elasticmedium, one along the x axis and the other along the y axis : xi _(1) = a cos ( omega t - kx), xi _(2) = a cos ( omega t - ky ) .Find the motion patternfo particles in the plane xy if both waves. (a) are tansverse and their oscillation directions coincide, (b) are longitudinal. |
| Answer» <html><body><p></p>Solution :`(a)` Equation of the resultant wave, <br/> `xi=xi_(1)+xi_(2)=2 a cos k ((y-x)/(2))<a href="https://interviewquestions.tuteehub.com/tag/xoa-3887560" style="font-weight:bold;" target="_blank" title="Click to know more about XOA">XOA</a> {omegat-(k(x+y))/(2)}, ` <br/> `=a'cos { omegat-(k(x+y))/2}`,where `a^(') = 2 a cos k^(') ((y-x)/2)` <br/>Now, the equation of wave pattern is, <br/> `x+ y=k`, (aConst.) <br/> For sought plots see the answer `-` sheet of the problem book. <br/> For antinodes, i.e. <a href="https://interviewquestions.tuteehub.com/tag/maximum-556915" style="font-weight:bold;" target="_blank" title="Click to know more about MAXIMUM">MAXIMUM</a> intensity <br/> `cos((k(y-x))/(2))=+-1= cos n pi` <br/> or, `+-(x-y)=(2npi)/(k) = n lambda` <br/> or `y=x+- nlambda, n=0,1,2,.....` <br/> Hence, the particles of the medium at the <a href="https://interviewquestions.tuteehub.com/tag/points-1157347" style="font-weight:bold;" target="_blank" title="Click to know more about POINTS">POINTS</a>, lying ono the solide straight lines `(y=x+-n lambda)` , oscillate with maximum amplitude.ltbr. For nodes, i.e. minimum intensity, <br/> `cos ((k(y-x))/(2))=0` <br/> or `+- (k(y-x))/( 2)=(2n+1) ( pi)/(2)` <br/> or , `y=x+- ( 2n +1) lambda//2`, <br/> and hence the particles at the points, lying on dotted lines do not oscillate. <br/> `(b)` When the waves are longitudinal, <br/> For sought plots see the answer `-` sheet of the problem book. <br/> `k(y-x)=cos^(-1)((xi_(1))/(a))- cos^(-1)((xi_(2))/( a))` <br/> or, `(xi_(1))/( a)= cos { k(y-x)+cos ^(-1) ((xi_(2))/(a))}` <br/> `=(xi_(2))/( a) cos k(y-x) - <a href="https://interviewquestions.tuteehub.com/tag/sin-1208945" style="font-weight:bold;" target="_blank" title="Click to know more about SIN">SIN</a> k y( y-x) sin(cos ^(-1)((xi_(2))/(a)))` <br/> `=( xi_(2))/(a) cos k ( y-x)- sin k ( y-x) <a href="https://interviewquestions.tuteehub.com/tag/sqrt-1223129" style="font-weight:bold;" target="_blank" title="Click to know more about SQRT">SQRT</a>(1-(xi_(2)^(2))/( a^(2)))...(1)` <br/> from `(1)`,<br/> if `sin k ( y-x) = 0 =cos ( 2n +1) ( pi)/(2)` <br/> `(xi_(1)^(2))/(a)=1-xi_(2)^(2)//a^(2)`, acircle. <br/> Thus the particles, at the points, where `y= x+-(n+- 1//4) lambda` , will oscillate along circles, <br/> In general, all other particles will move along ellipse.</body></html> | |