Standing waves are produced by the superposition of two waves y 1 = 0.05 s i n ⁡ 3 π t - 2 x and y 2 = 0.05 sin ⁡ 3 π t + 2 x , where   x and y are in metre and t is in second. The amplitude of the particle at x = 0.5   m is [Given cos 57 . 3 ° = 0.54 ]

Step-by-step solution

Here, y 1 = 0.05 sin ⁡ ( 3 π t - 2 x ) y 2 = 0.05 sin ⁡ ( 3 π t + 2 x ) According to superposition principle, the resultant displacement is y = y 1 + y 2 = 0.05 sin ⁡ ( 3 π t - 2 x ) + sin ⁡ ( 3 π t + 2 x ) y = 0.05 × 2 sin ⁡ 3 π t cos ⁡ 2 x y = 0.01 cos ⁡ 2 x sin ⁡ 2 π t = R sin ⁡ 3 π t Where R = 0.1 cos ⁡ 2 x = amplitude of the resultant standing wave. At x = 0.5 m R = 0.1 cos ⁡ 2 x = 0.1 cos ⁡ 2 × 0.5 = 0.1 cos ⁡ 1 r a d i a n = 0.1 cos ⁡ 180 ° π = 0.1 cos ⁡ 57.3 ° R = 0.1 × 0.54 m = 0.054 m = 5.4 c m .