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Two waves travelling in opposite directions produce a standing wave. The individual wave functions are y_(1) = (4.0 cm) sin (3.0 x - 2.0t) y_(2) = (4.0 cm) sin (3.0 x + 2.0t) where x and y are in cm (a) Find the maximum displacement of a particle ofthe medium at x = 2.3 cm (b) Find the position of the nodes and antinodes. |
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Answer» Solution :(a) When the two waves are summed, the result is a standing wave whose mathematical representation is given by equation, with A = 4.0 CM and K = 3.0 RAD/cm`y = (2A sin kx) cos omegat = [(8.0 cm) sin (3.0 x)] cos (2.0t)` Thus, the maximum displacement of a particle at the position x = 2.3 cm is `y_(max) = [(8.0 cm) sin 3.0x]_(x = 2.3 cm)` ` = (8.0 m) sin (6.9 rad) = 4.6cm`(b) Because `k = 2PI //lambda = 3.0` rad/cm, we see that `lambda` ` = 2pi//3cm` Therefore, the antinodes are located at `x = n((pi)/(6.0))` cm (n = `1, 3, 5,.....) and the nodes are located at ` x = n (lambda)/(2) (pi)/(6.0)` cm (n = 1, 2, 3,......) |
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