Question

Two waves travelling in opposite directions produce standing waves. The individual wave functions are given by $ {y_1} = 4sin\left( {3x - 2t} \right) $ and $ {y_2} = 4sin\left( {3x + 2t} \right)cm $ , where $ x $ and $ y $ are in $ cm $ .
(A) The maximum displacement of the motion at $ x = 2.3cm $ is $ 4.63cm $ .
(B) The maximum displacement of the motion at $ t = 2.3s $ is $ 4.63cm $ .
(C) Nodes are formed at $ x $ values given by $ 0,\pi /3,2\pi /3,4\pi /3, \ldots $
(D) Antinodes are formed at $ x $ values given by $ \pi /6,\pi /2,5\pi /6,7\pi /6, \ldots . $

Answer 1

Hint: We are given with the equations of the wave and are also said that the waves are flowing in opposite directions. Thus, clearly their interaction will produce a stationary or a standing wave. Thus, we will firstly add the given waves algebraically. Then we will connect the answer with the options given.

Formulae Used: $ sinA + sinB = 2sin\left[ {\left( {A + B} \right)/2} \right]cos\left[ {\left( {A - B} \right)/2} \right] $

Complete Step By Step Solution
Given,
 $ {y_1} = 4sin\left( {3x - 2t} \right) $ And $ {y_2} = 4sin\left( {3x + 2t} \right) $
Now,
 $ {y_1} + {y_2} $ , we get
 $ y = 4sin\left( {3x - 2t} \right) + 4sin\left( {3x + 2t} \right) $
Taking $ 4 $ common, we get
 $ y = 4\left\{ {sin\left( {3x - 2t} \right) + sin\left( {3x + 2t} \right)} \right\} $
Then,
Applying laws of trigonometry, we get
 $ y = 4\left\{ {2sin\left( {3x - 2t + 3x + 2t} \right)/2\cos \left( {3x - 2t - 3x - 2t} \right)/2} \right\} $
After evaluation, we get
 $ y = 8sin3xcos\left( { - 2t} \right) $
But,
We know,
 $ cos\left( { - \theta } \right) = \theta $
Thus,
 $ y = 8\,sin3x\,cos2t $
Resultant amplitude, $ A = 8sin3x $
Now,
For $ x = 2.3cm $
 $ A = 8sin\left( {3 \times 2.3} \right) $
Thus, we get
 $ A = 4.63cm $
Now,
For nodes,
 $ A = 0 $
Thus, we can say
 $ sin3x = 0 $
Thus,
 $ 3x = 0,\pi ,2\pi ,3\pi ,4\pi , \ldots $
 $ \Rightarrow x = 0,\pi /3,2\pi /3,\pi ,4\pi /3, \ldots $
Antinodes are formed in t=between two nodes,
Thus,
For antinodes, x can take the values,
 $ x = \pi /6,\pi /2,5\pi /6,7\pi /6, \ldots $
Thus, The correct options are (A), (C), (D).

Additional Information
A standing wave is a wave where there is no amplitude at all. It is a resultant of two similar waves flowing in opposite directions.
Nodes are positions on a wave where the amplitude or the maximum displacement is always zero. In other words, we can say nodes are stationary positions on a wave.
Antinodes are positions where amplitude or the maximum displacement is to its extreme. In other words, we can say antinodes are positions on the wave which attains the value of the amplitude or which displaces to the maximum possible value in that case.

Note
From the formula or relation $ A = 8sin3x $ , we can say the maximum displacement or the amplitude is independent of time. Thus there is no meaning in considering (B) as it is showing that amplitude is dependent on time.