Problem 59 - Master Resource Book in JEE Main Physics

Question

Two \(\quad\) waves \(\quad y_{1}=A_{1} \sin \left(\omega t-\beta_{1}\right) \quad\) and \(y_{2}=A_{2} \sin \left(\omega t-\beta_{2}\right)\) superimpose to form a resultant wave whose amplitude is (a) \(\sqrt{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2} \cos \left(\beta_{1}-\beta_{2}\right)}\) (b) \(\sqrt{\left(A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2} \sin \left(\beta_{1}-\beta_{2}\right)\right.}\) (c) \(A_{1}+A_{2}\) (d) \(\left|A_{1}+A_{2}\right|\)

Step-by-Step Solution

Answer

(a) \(\sqrt{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2} \cos(\beta_{1}-\beta_{2})}\) is the correct answer.

Step 1: Understanding Superposition of Waves

When two waves combine, the resulting wave is the sum of the individual waves. Mathematically, this is expressed as: \[ y_{r} = y_{1} + y_{2} = A_{1} \sin(\omega t - \beta_{1}) + A_{2} \sin(\omega t - \beta_{2}) \]

Step 2: Using Trigonometric Identities

To find the amplitude of the resultant wave, we can use the trigonometric identity for the sum of sines: \[ \sin C + \sin D = 2 \sin \left( \frac{C + D}{2} \right) \cos \left( \frac{C - D}{2} \right) \] This will help us rewrite the sum of the individual wave functions.

Step 3: Expressing Resultant Amplitude

Substitute the expressions for \(y_1\) and \(y_2\) using the identity:\[ A_{r} \sin \left(\omega t - \beta \right) = \sqrt{A_{1}^{2}+A_{2}^2 + 2 A_{1} A_{2} \cos(\beta_1 - \beta_2)} \sin \left(\omega t - \beta\right) \] This shows that the amplitude of the resultant wave is given by the formula involving the cosine of the phase difference.

Step 4: Selecting the Correct Option

The amplitude derived from the formula is given by: \[ \sqrt{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2} \cos(\beta_{1} - \beta_{2})} \] This matches option (a) in the problem statement.

Key Concepts

Trigonometric Identities

When dealing with wave interference or superposition, trigonometric identities become extremely helpful. This is because they allow us to simplify expressions and solve equations that involve trigonometric functions like sine and cosine. One key identity used in wave superposition is the sum identity for sine:

• \( \sin C + \sin D = 2 \sin \left( \frac{C + D}{2} \right) \cos \left( \frac{C - D}{2} \right) \)

This identity allows us to express the sum of two sine functions into a product form, which is easier to handle and manipulate mathematically.Applying this to our wave functions \(y_1\) and \(y_2\), we transform \(A_{1} \sin(\omega t - \beta_{1}) + A_{2} \sin(\omega t - \beta_{2})\) into a more manageable form. This transformation is vital to identifying the resultant amplitude and understanding the wave's behavior through the interference process.

Resultant Amplitude

In wave superposition, understanding the "resultant amplitude" is crucial. The resultant amplitude is the amplitude of the wave formed when two or more waves overlap. It's not just a simple addition of the individual amplitudes. Instead, it's calculated based on both the amplitudes and phase difference of the combining waves.Consider two waves given by the functions \(y_1 = A_1 \sin(\omega t - \beta_1)\) and \(y_2 = A_2 \sin(\omega t - \beta_2)\). These waves combine to form a new wave with its own distinct amplitude, defined by:

• \[ A_{r} = \sqrt{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2} \cos(\beta_1 - \beta_2)} \]

This formula shows that the resultant amplitude depends on the magnitudes of the individual amplitudes \(A_1\) and \(A_2\), as well as the cosine of the phase difference \(\beta_1 - \beta_2\), which reflects how in-phase or out-of-phase the waves are.

Phase Difference

The phase difference between waves is a key concept in understanding wave interference and superposition. It determines how the waves interact with each other—either constructively or destructively.The phase difference \(\beta_1 - \beta_2\) is the difference in phase angles of the two waves, \(y_1\) and \(y_2\). This difference can shift the way these waves add together.When using the resultant amplitude formula \(\sqrt{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2} \cos(\beta_{1} - \beta_{2})} \), the expression \(\cos(\beta_1 - \beta_2)\) is critical.

• If this cosine term is close to 1, the waves are in-phase, leading to constructive interference and a large resultant amplitude.
• If the cosine term is close to -1, the waves are out-of-phase, leading to destructive interference and a smaller resultant amplitude.
• A cosine term of 0 indicates the waves are perfectly out-of-phase, potentially canceling each other, depending on their amplitudes.

Understanding phase differences can thus help predict the resultant wave pattern from wave superpositions.