Problem 25 - Master Resource Book in JEE Main Physics

Question

Two coherent waves are represented by \(y_{1}=a_{1} \cos \omega t\) and \(y_{2}=a_{2} \sin \omega t\), superimposed on each other. The resultant intensity is proportional to (a) \(\left(a_{1}+a_{2}\right)\) (b) \(\left(a_{1}-a_{2}\right)\) (c) \(\left(a_{1}^{2}+a_{2}^{2}\right)\) (d) \(\left(a_{1}^{2}-a_{2}^{2}\right)\)

Step-by-Step Solution

Answer

The resultant intensity is proportional to \(a_1^2 + a_2^2\), matching option (c).

Step 1: Write the expression for the resultant wave

The resultant wave from the superposition of the two given waves is obtained by adding them algebraically: \[ y = y_1 + y_2 = a_1 \cos \omega t + a_2 \sin \omega t. \]

Step 2: Express the resultant in a trigonometric form

To simplify and find the resultant intensity, express the resultant wave in the form \( R \sin(\omega t + \phi) \) or \( R \cos(\omega t + \phi) \). We can use the identity for a linear combination of sine and cosine. This can be rewritten as \[ y = R \cos(\omega t + \phi), \]where \[ R = \sqrt{a_1^2 + a_2^2}. \]

Step 3: Calculate the resultant intensity

Intensity is proportional to the square of the amplitude. Hence, the intensity \(I\) will be proportional to \[ I \propto R^2 = a_1^2 + a_2^2. \]

Step 4: Match with given options

Compare the expression for the resultant intensity with the given options. We find that the intensity expression \( a_1^2 + a_2^2 \) matches with option (c).

Key Concepts

Coherent Waves

Coherent waves are waves that have a constant phase relationship. This means that the peaks and troughs of these waves align in a predictable way over time. Coherent waves are essential for creating clear interference patterns, which occur during wave superposition.
In the given exercise, two coherent waves are represented by their respective equations:

• \( y_{1} = a_{1} \cos \omega t \)
• \( y_{2} = a_{2} \sin \omega t \)

These two waves maintain a stable phase relationship due to their coherent nature. Understanding this relationship allows us to predict the resultant wave when they are combined or superimposed.

Resultant Intensity

The resultant intensity of the superposition is a measure of the energy carried by the combined wave. This intensity is closely related to the amplitude of the resultant wave. When two waves interfere, the intensity depends on how their amplitudes combine.
In this problem, the key to finding the resultant intensity is remembering that intensity is proportional to the square of the wave's amplitude. As we calculate the resultant wave's amplitude to be \( R = \sqrt{a_{1}^2 + a_{2}^2} \), we find the resultant intensity as:

• \( I \propto R^2 = a_{1}^2 + a_{2}^2 \)

This means that the correct answer according to the given options is the one corresponding to \( a_1^2 + a_2^2 \). This demonstrates how resultant intensity provides insight into how energy combines in coherent wave interactions.

Trigonometric Form

The trigonometric form of a wave is a useful way of expressing complex wave interactions. When we have combinations of sine and cosine functions, converting them to a single trigonometric form simplifies calculations.
In this exercise, we take advantage of this by rewriting the sum of the two coherent waves:

• \( y = a_{1} \cos \omega t + a_{2} \sin \omega t \)

as a single wave function

• \( y = R \cos(\omega t + \phi) \)

where \( R \) is the amplitude we just calculated. This transformation uses the identity for linear combinations of sine and cosine, which makes it easier to calculate and understand the resultant wave's behavior.

Amplitude Calculation

The amplitude of a wave is a critical factor in determining its intensity and the overall energy it carries. In exercises dealing with coherent waves, calculating the amplitude of the resultant wave is key for understanding its intensity.
For this problem, the amplitude \( R \) is calculated using the Pythagorean identity:

• \( R = \sqrt{a_{1}^2 + a_{2}^2} \)

This equation stems from the way sine and cosine components combine in trigonometric functions. The amplitude \( R \) effectively incorporates both components, providing a unified parameter that reflects the maximum displacement of the wave.
Thus, understanding amplitude calculation is critical for solving problems related to wave superposition and interference.