Understanding Wave Interference and Phase Difference

This problem involves the superposition of two identical coherent waves. When waves superimpose, their resulting intensity depends on their phase difference. We are given the relationship between the actual resultant intensity and the maximum possible resultant intensity ($I_{max}$) and asked to find the specific phase difference ($\phi$) at the point of superposition.

Key Concepts for Wave Interference

• Coherent Waves: Waves having the same frequency and a constant phase difference.
• Superposition Principle: When two or more waves overlap, the resultant displacement at any point is the vector sum of the displacements due to individual waves.
• Intensity: Intensity ($I$) of a wave is proportional to the square of its amplitude ($A$), i.e., $I \propto A^2$.
• Maximum Intensity ($I_{max}$): Occurs when waves interfere constructively (phase difference is $0, 2\pi, 4\pi, ...$). The amplitude is the sum of individual amplitudes.
• Resultant Intensity: For two waves with amplitudes $A_1$ and $A_2$ and phase difference $\phi$, the resultant amplitude $A$ is given by $A^2 = A_1^2 + A_2^2 + 2A_1A_2 \cos(\phi)$. The resultant intensity $I$ is proportional to $A^2$.

Step-by-Step Calculation of Phase Difference

1. Define Individual Wave Properties: Since the two waves are identical, let their amplitudes be $A_0$. The intensity of each wave is $I_0 = kA_0^2$ for some constant $k$.

2. Calculate Maximum Intensity ($I_{max}$): The maximum possible resultant intensity occurs during constructive interference when the phase difference $\phi = 0$. In this case, the amplitudes add up directly.

   Resultant amplitude $A_{max} = A_0 + A_0 = 2A_0$.

   The maximum intensity is $I_{max} = k(A_{max})^2 = k(2A_0)^2 = k(4A_0^2)$.

3. Express General Resultant Intensity ($I$): For any phase difference $\phi$ between the two identical waves, the resultant amplitude $A$ is:

   $A^2 = A_0^2 + A_0^2 + 2A_0A_0 \cos(\phi) = 2A_0^2 + 2A_0^2 \cos(\phi) = 2A_0^2 (1 + \cos(\phi))$

   The resultant intensity $I$ is:

   $I = kA^2 = k[2A_0^2 (1 + \cos(\phi))]$

4. Use the Given Intensity Relationship: We are told that the resultant intensity at the point is $I = \frac{I_{max}}{4}$.

   Substituting the expression for $I_{max}$:

   $I = \frac{1}{4} [k(4A_0^2)] = kA_0^2$.

5. Equate Intensity Expressions and Solve for $\cos(\phi)$:** Now, we equate the general expression for $I$ with the specific value $kA_0^2$:

   $kA_0^2 = k[2A_0^2 (1 + \cos(\phi))]$

   Divide both sides by $kA_0^2$ (since $A_0 \neq 0$):

   $1 = 2 (1 + \cos(\phi))$

   Divide by 2:

   $\frac{1}{2} = 1 + \cos(\phi)$

   Rearrange to find $\cos(\phi)$:

   $\cos(\phi) = \frac{1}{2} - 1 = -\frac{1}{2}$

6. Determine the Phase Difference ($\phi$): We need to find the phase difference $\phi$ such that $\cos(\phi) = -\frac{1}{2}$.

   The principal values for $\phi$ in the range $[0, 2\pi]$ that satisfy this condition are $\phi = \frac{2\pi}{3}$ and $\phi = \frac{4\pi}{3}$.

   Looking at the options provided, $\frac{2\pi}{3}$ is listed.

Conclusion

Based on the calculation, the phase difference between the two identical coherent waves when the resultant intensity is one-fourth of the maximum possible resultant intensity is $\frac{2\pi}{3}$.