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Consider a string fixed at one end . A travelling wave given by the wave equationy = A sin ( omega t - k x) is incident on it . ltbr gt Show that at the fixed end of a string the waves suffers a phase change of pi, i.e., as it travels back as if the wave isinverted. |
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Answer» Solution :Wave equation of incident wave is `y_(1) A sin ( omega t - kx) ( "positive x - direction")` When the ave strikes the fixed end it must be reflected. The wave will now be travelling in the negativex - direction , and so its equation is ` y_(2) A sin ( omega t + kx + PHI ) ( "negative x - direction")` The phase constant has been added to account for any phase change after reflection . Let us take ` x = 0 ` at the fixed end . The behaviour of a wave at particularpositions is governed by appropriate boundary conditions . For fixed ends , the boundary CONDITION is that the end point is a node . The resultant motion due to incident and reflected wave is ` y= y_(1) + y_(2)` `y = A [ sin ( omega t - kx) + sin ( omega t + kx + phi)]`(i) The boundary condition that must be satisfiedby the resultant wave on string is ` y = 0 at x = 0 ` On substituting these values in Eq. (i) , we obtain `sin omega t + sin ( omega t + phi ) = 0 ` `sin omega t = - sin ( omega t + phi)`(ii) Equation (ii) must be satisfied at all times . For the sake of convenience , we take ` omega t = pi //2`. `sin ((pi)/(2) + phi)= -1` `sin theta = -1 ` implies `theta` can have any of the following values. `3 pi//2 , 7 pi //2 , 11 pi//2 ,....` Therefore `phi` can have any of the values `pi , 3 pi , 5 pi` , etc. All these values are PHYSICALLY possible and distinguishable . We CHOOSE the simplest one , so `phi = pi`. |
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