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Derive an expression for a one - dimensional simple harmonic progressive wave travelling in the direction of thepositive x - axis. Express it in terms ofA, lambda, v , t and x . |
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Answer» Solution :Consider a simple harmonic progressive wave travelling with a speed valong the positive x-axis. Let y denote the displacement of a particle of themedium from its mean position. We assume that theinitial phase of the oscillatory motion is zero . At time t, the displacement of the particle of the medium situated at the origin is ` y = A sin omega t`...(1) where A is the amplitude of the wave and `omega` is related to thefrequency n of the wave motion by theequation `omega = 2 pi n`.  Simple harmonic progressive wave Consider a particle of the medium situated at point P at a distance x from the origin O as shown in thefigure. This particle also PERFORMS SHM with the same amplitude A and thefrequency n.However, thedisturbance at O reaches P only after some time , i.e., the particle at P displacement of the particle at P is ` y = A sin ( omega t - alpha)`...(2) Since a PATH difference of `lambda` (wavelength) corresponds to a phase defferenceof ` 2PI ` radians , a path difference of x corresponds to a phase difference of ` alpha` such that ` lambda/(2pi) = x/alpha . :.alpha = (2 pi)/lambda x`. `:. y = A sin (OMEGAT - (2pi)/lambda x)`....(3) If n is the frequency of vibration , `omega = 2 pi n = (2 pi v)/lambda`...(4) where v is the speed of the wave, ` :.y = A sin 2 pi (nt - x/lambda)` ...(5) Equation (5) is the equation of a simple harmonic progressive wave travelling in thepositive direction of the x - axis. Also from Eq. (4), `y = A sin 2 pi (v/lambda t - x/lambda)` ` :.y = A sin (2 pi)/lambda (vt - x)`....(6) in terms of v and `lambda` , as required. |
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