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Question : 13 of 27
Marks: +1, -0
Given below are some functions of x and t to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all :
(a) y=2cos⁡(3x)sin⁡(10t)y = 2 \cos (3x) \sin (10t)
(b) y=2x−vty = 2 \sqrt{x - v t}
(c) y=3sin⁡(5x−0.5t)+4cos⁡(5x−0.5t)y = 3 \sin (5x - 0.5 t) + 4 \cos (5x - 0.5t)
(d) y=cos⁡xsin⁡t+cos⁡2xsin⁡2ty = \cos x \sin t + \cos 2x \sin 2t
Solution:  
(a) This equation has two harmonic functions of each x and t separately, so it represents stationary wave.
(b) This function does not represent any wave as it contains no harmonic function.
(c) It represents progressive/travelling harmonic wave as the arguments of cosine and sin functions are same.
(d) This equation is the sum of two functions cos⁡xsin⁡t\cos x \sin t and cos⁡2xsin⁡2t\cos 2x \sin 2t each representing a stationary wave. Therefore it represents superposition of two stationary waves.
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