A simple harmonic progressive wave is represented by the equation: \[y = 8\sin 2\pi \left( {0.1x - 2t} \right)\] where \[x\] and \[y\] are in cm and \[t\] is in seconds. At any instant the phase difference between two particles separated by \[{\text{2}}{\text{.0}}\,{\text{cm}}\]in the \[x\] direction is
A. \[{18^ \circ }\]
B. \[{36^ \circ }\]
C. \[{54^ \circ }\]
D. \[{72^ \circ }\]

Hint: Compare the given equation with the generalized equation of simple harmonic wave and then find out the value of propagation constant of the given simple harmonic wave. Use the value of propagation constant to find out the phase difference between the particles.

Complete step by step answer:
Given, the wave equation of a simple harmonic progressive wave is
\[y = 8\sin 2\pi \left( {0.1x - 2t} \right)\]
\[ \Rightarrow y = 8\sin \left( {0.2\pi x - 4\pi t} \right)\] …………………..(1)
And the distance between the two particles is \[\Delta x = 2.0\,{\text{cm}}\] …………………...(2)

The generalized equation for a simple harmonic wave travelling along x-axis is written as,
\[y = A\sin \left( {kx - wt} \right)\] …………………………….(3)
where \[A\]is the amplitude of the wave, \[k\]is propagation constant, \[w\] is the angular frequency, \[t\]is the time and \[x\] is the displacement of the particle.
Comparing equation (1) and (2), we get the propagation constant to be
\[k = 0.2\pi \] ………………………………...(4)
The formula for propagation constant is,
\[k = \dfrac{{2\pi }}{\lambda }\] ………………………………...(5)
Where \[\lambda \] is the wavelength of the wave.
Equating equations (3) and (4), we get
\[0.2\pi = \dfrac{{2\pi }}{\lambda }\]
\[ \Rightarrow \lambda = \dfrac{{2\pi }}{{0.2\pi }} = 10\,{\text{cm}}\] ……………………………….(6)
The formula to find out the phase difference between two particles is,
\[\Delta \phi {\text{ = }}\dfrac{{2\pi }}{\lambda }\Delta x\] …………………………………..(7)
where \[\Delta x\] is the distance between two particles and \[\lambda \] is the wavelength of the wave.
Now, putting the values of \[\lambda \] and \[\Delta x\] from equation (6) and (2) respectively, in equation (7), we get

\[ \Rightarrow \Delta \phi = \dfrac{{2\pi }}{{10}} \times 2 \\
\Rightarrow \Delta \phi = \dfrac{{2\pi }}{5} = {72^ \circ } \\\]

Therefore, the phase difference between two particles separated by a distance \[\Delta x = 2.0\,{\text{cm}}\] along x-axis direction is \[{72^ \circ }\]

So, the correct answer is “Option D”.

Note:
While comparing the generalized equation with the given equation, we should carefully check for the values of propagation constant and angular frequency. For example in this question \[2\pi \] was given outside of the bracket and if we don’t consider this factor and take \[k\] as \[0.1\] then it would lead us to the wrong answer.