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

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\left ( 5x-\frac{t}{2} \right )+ 3 \cos \left ( 5x-\frac{t}{2} \right )

c) y = 10 \cos [(252-250)\pi t] \cos [(252+250) \pi t]

d) y = 100 \cos (100 \pi t + 0.5x)

State which of these represent

a) a travelling wave along the -x direction

b) a stationary wave

c) beats

d) a travelling wave along +x direction

Give reasons for your answers

Answers (1)

(a) A wave travelling along (-x) direction has +kx as in

 d) y = 100 \cos (100 \pi t + 0.5x)

(b ) A stationary wave equation is

 a) y = 5 \cos (4x) \sin (20t)

  1. Beats involved and \left (v_{1}-v_{2} \right ) So they can be represented by

y=\left [\cos\left (252-250 \right )\pi t \right ] i.e. option c

 

  1. y = 4 \sin\left ( 5x-\frac{t}{2} \right )+ 3 \cos \left ( 5x-\frac{t}{2} \right )

Let 4=a\cos\phi and 3=a\sin\phi ;

\tan\phi =\frac{3}{4} or \phi = tan^{-1} \frac{3}{4}

a^{2}\cos \phi ^{2}+a^{2}\sin \phi ^{2}=3^{2}+4^{2}  

a^{2}(\cos^{2}\phi +sin^{2}\phi )=9+16

 a^{2}=25 

a=5

 Now,

y=a\cos \phi \sin\left (5x-\frac{t}{2} \right )+a\sin \phi \cos\left (5x-\frac{t}{2} \right )

 y = a \sin \left (5x -\frac{t}{2} + \phi \right )

y = 5 \sin \left (5x -\frac{t}{2} + \phi \right )

Which represents the progressive wave in +x direction as the sign of kx or (5x) and\omega t\left ( \frac{1}{2t} \right ) are opposite so it travels in+x direction.

Posted by

infoexpert24

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