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Asked by sarveshvibrantacademy | 07 Apr, 2019, 11:24: AM
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Figure shows the longitudinal displacement variation of air particles along the wave propagation direction (x-axis).
Corresponding pressure variation also is shown in figure.
 
(A) |∂ξ /∂x|C = |∂ξ /∂x|R .  This is true.  At maximum compression point the slope is decreasing. At maximum rarifaction point slope is increasing.
                                            But the magnitude of rate of change of displacement with respect to distance is same. This can be seen from figure.
 
(B) |∂ξ /∂t|C = |∂ξ /∂t|R = 0 . This is true. At the compression and rarifaction points, displacement of air particles is almost nil. 
                                               Hence time variation of displacement at these points also zero.
 
(C) PC - PR = 2 |∂ξ /∂x|C × Bulk-Modulus . This true as per the explanation given below
                         Let  PC - PR = ΔP  , change in pressure at compression point C.
                         
                         2 ∂ξ is the net displacement of air particles around compression point C.
                         if we consider an area ΔA perpendicular to the longitudinal motion,
                          then (2 ∂ξ )ΔA is change in volume ΔV.   original volume of air particles is ∂xΔA

                          Hence begin mathsize 12px style fraction numerator P subscript C minus P subscript R over denominator 2 open vertical bar begin display style fraction numerator partial differential xi over denominator partial differential x end fraction end style close vertical bar subscript C end fraction space equals space fraction numerator increment P over denominator begin display style fraction numerator 2 partial differential xi space increment A over denominator partial differential x space increment A end fraction end style end fraction space equals space fraction numerator increment P over denominator begin display style fraction numerator increment V over denominator V end fraction end style end fraction space equals space B u l k space m o d u l u s end style
 
 
(D) Particles of air are stationary between C and R - False
       Maximum displacement occurs at mid point of C and R. Hence particles of air are not stationary
Answered by Thiyagarajan K | 07 Apr, 2019, 21:05: PM

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