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Question

Answers

(A) $2\pi $

(B) $\dfrac{\pi }{2}$

(C) $\dfrac{\pi }{4}$

(D) $\pi $

Answer
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We should know that the time period of the wave is given as t. The value of t is 0.04 seconds.

Now the velocity of propagation of oscillation is given as v. The value of v is 300 m/ sec.

So now the wavelength of the oscillation is given as $\lambda $. The value of $\lambda $ is given as $\lambda = v \times t$.

Now we have to substitute the values from the question to the given expressions.

So the value of wavelength or $\lambda $becomes :

$

\lambda = 300 \times 0.04 \\

\Rightarrow \lambda = 12m \\

$

Now we have to use the relation between the path difference that is $\Delta x$ and the phase difference, which is given below as:

$\dfrac{{\Delta x}}{\lambda } = \dfrac{{\Delta \phi }}{{2\pi }}$

Now it is known that we have to find the phase difference at 10 m from the source of oscillation. So given below are the expressions:

$\Delta x = 10$

So we can write that:

$

\dfrac{{10}}{{12}} = \dfrac{{\Delta \phi }}{{2\pi }} \\

\Rightarrow \dfrac{{10}}{{12}} \times 2\pi = \Delta \phi \\

\Rightarrow \Delta \phi = \dfrac{{5\pi }}{3} \\

$

Now we have to find the phase difference at 16 m from the source of the oscillation. So the expressions are as follows:

$\Delta x = 16$

So we can write that:

$

\dfrac{{16}}{{12}} = \dfrac{{\Delta \phi }}{{2\pi }} \\

\Rightarrow \dfrac{{16}}{{12}} \times 2\pi = \Delta \phi \\

\Rightarrow \Delta \varphi = \dfrac{{8\pi }}{3} \\

$

Now we have to find the phase difference at 16 m and 10 m from the source. So the expressions are as follows:

$

\Delta {\phi _{16}} - \Delta {\phi _{10}} = \dfrac{{8\pi }}{3} - \dfrac{{5\pi }}{3} \\

\Rightarrow \Delta {\phi _{16}} - \Delta {\phi _{10}} = \dfrac{{3\pi }}{3} \\

\Rightarrow \Delta {\phi _{16}} - \Delta {\phi _{10}} = \pi \\

$

So we can say that the difference of phases between the oscillations of two points at distances 10 and 16 m respectively from the source of oscillation is $\pi $.

We have also come across the term wavelength. By wavelength we mean the length of one wave or we can say the distance from a specific point on one wave to the same point in case of another wave.

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