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Chapter 9: Problem 29

The two waves having intensities in the ratio \(1: 9\) produce interference. The ratio of the maximum to the minimum intensities is equal to (A) \(10: 8\) (B) \(9: 1 \quad\) (C) \(4: 1\) (D) \(2: 1\)

Short Answer

Expert verified
The ratio of the maximum to the minimum intensities when two waves with intensities in the ratio 1:9 interfere is 10:8. \( \boxed{\text{(A) }10:8} \).

Step by step solution

01

Represent intensities of the two waves

: Let the intensity of the first wave be I鈧� and that of the second wave be I鈧�. We are given that the ratio of their intensities is 1:9, which can be written as: I鈧� : I鈧� = 1 : 9
02

Use the formula for intensity

: The intensity of a wave is given by the formula: \[I = A^2\] where A is the amplitude of the wave. From the given ratio, we can associate the amplitudes of the two waves as A鈧� and A鈧� with their intensities: \[I鈧� = A鈧乛2\] \[I鈧� = A鈧俕2\]
03

Find the amplitude ratio

: Using the given intensity ratio and the intensity formula, we can relate the amplitude ratio to the intensity ratio: \[\frac{I鈧亇{I鈧倉 = \frac{A鈧乛2}{A鈧俕2} \Rightarrow \frac{1}{9} = \frac{A鈧乛2}{A鈧俕2}\]
04

Calculate the resultant amplitude

: When two waves interfere, their amplitudes add up according to the principle of superposition. The maximum and minimum resultant amplitudes occur when the two waves are in phase (constructive interference) and out of phase (destructive interference), respectively. For constructive interference (maximum amplitude): \[A_{max} = A鈧� + A鈧俓] For destructive interference (minimum amplitude): \[A_{min} = |A鈧� - A鈧倈\]
05

Calculate the maximum and minimum intensities

: Using the formula for intensity, we can find the maximum and minimum intensities based on the maximum and minimum amplitudes: \[I_{max} = A_{max}^2 = (A鈧� + A鈧�)^2\] \[I_{min} = A_{min}^2 = (A鈧� - A鈧�)^2\]
06

Find the ratio of the maximum to the minimum intensities

: Finally, we need to find the ratio of the maximum to the minimum intensities. We can use algebraic manipulation to find the ratio: \[\frac{I_{max}}{I_{min}} = \frac{(A鈧� + A鈧�)^2}{(A鈧� - A鈧�)^2}\] Using the amplitude ratio found in Step 3 and some algebraic simplification, \[\frac{(A鈧� + A鈧�)^2}{(A鈧� - A鈧�)^2} = \frac{(A鈧乛2 + 2 A鈧丄鈧� + A鈧俕2)}{(A鈧乛2 - 2 A鈧丄鈧� + A鈧俕2)} = \frac{10}{8}\] So, the ratio of the maximum to the minimum intensities is 10:8, which is the same as option (A).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Superposition Principle
The superposition principle is a fundamental concept in physics that defines how two or more waves can interact with each other. When two waves meet, they overlap, and the resultant wave at any point is the algebraic sum of the amplitudes of the individual waves at that point. This interaction remains true for waves of light, sound, and water, among others.

One way to visualize this is by imagining two ripples crossing paths on a water surface. Where they meet, they either create a higher wave (if the peaks coincide) or cancel each other out (if a peak meets a trough). To fully grasp this principle, it's crucial to understand that the waves don't alter each other permanently; rather, the resulting wave formation is transient, lasting only as long as the waves are intersecting.
Constructive and Destructive Interference
Constructive and destructive interference are two outcomes of the superposition principle. When two waves are in phase, meaning their peaks (or troughs) align perfectly, they reinforce each other, leading to a higher amplitude. This effect is known as constructive interference.

Conversely, when two waves are out of phase, such that the peak of one aligns with the trough of another, they partially or completely cancel each other out, resulting in a lower amplitude or even flat line in the case of total cancellation. This phenomenon is known as destructive interference.

The textbook solution refers to these two types of interference when calculating the maximum (constructive) and minimum (destructive) intensities of waves. The maximum amplitude occurs when waves add up coherently and the minimum amplitude is the absolute difference in their amplitudes, reflecting constructive and destructive interference respectively.
Amplitude and Intensity Relationship
The relationship between amplitude and intensity is an essential aspect of understanding wave behavior. Intensity measures how much energy a wave transmits per unit area per unit time and is proportional to the square of the wave's amplitude. In a formula, this relationship is expressed as:
\[I = A^2\]
where \(I\) stands for intensity and \(A\) represents amplitude. When dealing with interference, if one understands this relationship, then it becomes clear how the intensities will differ for the maximum and minimum amplitudes caused by constructive and destructive interference.

In the exercise, by knowing the amplitudes of two waves and their respective intensities, we can predict the change in intensity that results from their interference. The square dependence means that even small changes in amplitude can lead to significant changes in intensity, which is crucial while calculating the ratio of the maximum to minimum intensities for the problem at hand.

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Most popular questions from this chapter

A particle moves on the \(x\)-axis as per the equation \(x=x_{0} \sin ^{2} \omega t .\) The motion is simple harmonic (A) With amplitude \(x_{0}\) (B) With amplitude \(2 x_{0}\) (C) With time period \(\frac{2 \pi}{\omega}\) (D) With time period \(\frac{\pi}{\omega}\)

In forced oscillation of a particle the amplitude is maximum for a frequency \(\omega_{1}\) of the force while the energy is maximum for a frequency \(\omega_{2}\) of the force; then (A) \(\omega_{1}<\omega_{2}\) when damping is small and \(\omega_{1}>\omega_{2}\) when damping is large (B) \(\omega_{1}>\omega_{2}\) (C) \(\omega_{1}=\omega_{2}\) (D) \(\omega_{1}<\omega_{2}\)

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