Q34 of 36 Page 110

Given below are some functions of x and t to represent the displacement of an elastic wave.

(a) y = 5 cos (4x) sin (20t)


(b) y = 4 sin (5x – t/2) + 3 cos (5x – t/2)


(c) y = 10 cos [(252 – 250) πt ] cos [(252+250)πt ]


(d) y = 100 cos (100πt + 0.5x)


State which of these represent


(a) a travelling wave along –x direction


(b) a stationary wave


(c) beats


(d) a travelling wave along +x direction.


Given reasons for your answers.


(a) The given equation is



by using the property



The given equation can be represented as,



which is the superposition of the same progressive wave in the opposite direction and therefore is the equation of a stationery wave.


(b) The given equation represents a travelling wave along the +x direction as the equation has terms of (ωt - kx) which cannot be separated.


(c) the given equation is



using the property



The given equation can be represented as,



which is the superposition of two wave having very similar frequencies of 504 Hz and 500 Hz which constitute beats.


(d) The given equation represents a travelling wave along the -x direction as the equation has terms of (ωt + kx) which cannot be separated.


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In the given progressive wave y = 5 sin (100πt – 0.4πx) where y and x are in m, t is in s. What is the

(a) amplitude


(b) wave length


(c) frequency


(d) wave velocity


(e) particle velocity amplitude.


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For the harmonic travelling wave y = 2 cos 2π (10t–0.0080x + 3.5) where x and y are in cm and t is second. What is the phase difference between the oscillatory motion at two points separated by a distance of,

(a) 4 m


(b) 0.5 m


(c)


(d) (at a given instant of time)


(e) What is the phase difference between the oscillation of a particle located at x = 100cm, at t = T s and t = 5 s?