Two traveling waves are generated on the same taut stnng. Individually, the two

Question

Two traveling waves are generated on the same taut stnng. Individually, the two traveling waves can be described by the following two equations: y_1(x, t) = (3.41 cm) sin(k_1x + (0.208 rad/s)t + phi_1) y_2(x, t) = (5.03 cm) sin(k_2x minus (10.2 rad/s)t + phi_2) If both of the above traveling waves exist on the string at the same time, what is the maximum positive displacement that a point on the string can ever have? What are the smallest positive values of the unknown phase conslanls (in radians) such that the above displacement occurs at the origin (x = 0) at time t = 1.68 s?

Explanation / Answer

Maximum displacement will be when both wave are in phase and Y1 and Y2 both are maximum.

then Y = Y1 + Y2 = 3.41 + 5.03 = 8.44 cm . ...........Ans

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so sin(k1x + 0.208t + phi1) = sin(k2x - 10.2t + phi2) = 1

at x = 0 and t = 1.68

sin(0.208x1.68 + phi1) = sin(0 - 10.2x1.68 + phi) = 1

sin(0.35 + phi1) = 1

phi1 = pi/2 - 0.35 = 1.22 rad ......Ans

sin(-17.14 + phi2) = 1

- 17.14 + phi2 = (2n + 1) pi /2

for smallaest phi2, n = -5

phi2 = - 9 pi/2 + 17.14 = 3 rad

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