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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 $\pi /6,\pi /2,5\pi /6,7\pi /6, \ldots .$

Last updated date: 13th Jun 2024
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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).

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.