Question

The best combination of independent sources to produce sustained pattern among the following is:
$1. \quad { Y }_{ 1 }=\ asin\left( \omega t \right)$
$2. \quad { Y }_{ 2 } =\ acos\left( \omega t \right)$
$3. \quad { Y }_{ 3 } =\ asin\left( \omega t+\dfrac { \pi }{ 4 } \right)$
$4. \quad { Y }_{ 4 }=\ 2asin \left( \omega t + \pi \right)$
$\ A. \quad Y_{1},Y_{2}\ only$
$\ \ B. \quad Y_{2}, \ Y_{3} \ only$
$\ C. \quad Y_{3},\ Y_{4} \ only$
$\ \ D. \quad None \ of \ these$

Answer 1

Hint: The main conditions for sustained pattern or sustained interference are that sources must be coherent. Their amplitude must not vary much. The phase difference between the sources must be zero and both the sources must be monochromatic. In addition to this, they both must be travelling in the same direction and both must be in the same plane.

Complete step-by-step answer:
Here we’ll use the concept of interference of light waves.
But first, let’s understand the standard wave equation.
$Y=asin\left( \omega t+\phi \right)$is called the standard wave equation.
Here ‘Y’ represents the displacement of wave particles at time ‘t’. Coefficient of trigonometric function ‘a’ is called the amplitude of the wave.’$\omega$’ is the angular frequency of the wave, which is the measure of angular displacement. ‘$\phi$’ is the initial phase difference of the wave. It is also called ‘epoch’.

Now, option A, B, C have the same amplitude. D has a different amplitude. Hence it can’t produce sustained interference with any other source.
A & B, both sources are in phase. C has huge phase difference which is non favorable for sustained interference. Hence it can’t produce sustained interference with any other source.
But A & B have a phase difference of $\ \dfrac \pi2$. Hence A & B also can’t produce sustained interference.
Hence the correct answer will be D. None of these.
Additional information: The daily life example of interference is thin film interference. When oil falls over a wet surface, it forms a thin layer. Oil having a different refractive index than that of water and air, produces interesting colored patterns. Also when we blow air in bubbles, it traps sunlight and produces the same phenomenon.

Note: Students must not be confused about the definition of coherent sources. Coherent sources means that the phase difference between the two sources must not change with time. Here we used the concept of interference of light. In many questions like these, students might have to proceed through options only and in some questions, it could be easy to proceed through options. Students must be ready both the ways.