Question
The particle displacement of a travelling longitudinal wave is represented by S = S (x, t). The midpoints of a compression zone and an adjacent rarefaction zone are represented by the letter ‘C’ and ‘R’. Which of the following is true?
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Solution
Sol. Let S = S0 sin (kx – wt) …. (1)
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So ∆p = $\frac{–\partial s}{\partial x}$ = – BkS0 cos (kx – wt) ….(2)
Form variation of excess pressure we see that point C is compression as ∆p is + ve maximum and points A & E are rare fraction as ∆p is – ve minimum.
(A) From s – x graph $\left| \frac{\partial s}{\partial x} \right|_{A} = \left| \frac{\partial s}{\partial x} \right|_{C}$
(B) Form equation (1) $\left| \frac{\partial s}{\partial t} \right|_{C} \neq \left| \frac{\partial s}{\partial t} \right|_{R}$
(C) PC – PR = 2∆Pmax = 2 BKS0 = 2B$\left| \frac{\partial s}{\partial x} \right|_{C}$
(D) $\frac{\partial s}{\partial x}$ = 0 at point B & D, midway b/w ‘C’ & ‘R’
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