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Chapter 12: Problem 56

When two waves of almost equal frequencies \(n_{1}\) and \(n_{2}\) are produced simultaneously, then the time interval between successive maxima is (a) \(\frac{1}{n_{1}-n_{2}}\) (b) \(\frac{1}{n_{1}}-\frac{1}{n_{2}}\) (c) \(\frac{1}{n_{1}}+\frac{1}{n_{2}}\) (d) \(\frac{1}{n_{1}+n_{2}}\)

Short Answer

Expert verified
The time interval between successive maxima is option (a) \(\frac{1}{n_{1}-n_{2}}\).

Step by step solution

01

Understanding the Problem

We need to determine the time interval between successive maxima when two waves with frequencies \(n_1\) and \(n_2\) interfere. This involves finding how often the wave interference reaches a peak, or maximum.
02

Concept of Beat Frequency

When two waves of slightly different frequencies interfere, they create a beat frequency, \(f_{beat}\), which is the difference of the two frequencies: \(f_{beat} = |n_1 - n_2|\). The beat frequency tells us how often the maxima (or beats) occur.
03

Finding Time Interval for Maxima

To find the time interval between successive maxima, we use the relationship, \(T = \frac{1}{f_{beat}}\), where \(T\) is the period of the beat frequency. Since \(f_{beat} = |n_1 - n_2|\), the time interval between successive maxima is \(T = \frac{1}{|n_1 - n_2|}\).
04

Matching the Solution with Given Options

Now, let's compare our derived expression \(\frac{1}{|n_1 - n_2|}\) with the answer options given. The correct match is option (a) \(\frac{1}{n_1 - n_2}\), assuming that \(n_1 > n_2\), so the absolute value is not necessary.

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Key Concepts

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

Beat Frequency
When two sound waves of slightly different frequencies collide, a fascinating phenomenon occurs known as beat frequency. Think of it like two singers harmonizing their voices but being slightly off. The combined sound results in alternating loud and soft sounds, often referred to as "beats."

The beat frequency, denoted as \(f_{beat}\), is simply the absolute difference between the frequencies of the two waves, calculated using the formula:
  • \(f_{beat} = |n_1 - n_2|\)
This value tells us how frequently the loud beats happen. If the two frequencies are close, the beats occur slowly, but as the frequency difference increases, the beats become more rapid. Understanding beat frequency is essential in music and acoustics to grasp how sound waves interact in our everyday environment.
Wave Interference
Wave interference happens when two or more waves pass over each other and combine. There are two main types of interference: constructive and destructive.

  • Constructive Interference: Occurs when waves align in phase, meaning the crest of one wave meets the crest of another. This alignment amplifies the overall wave, creating a larger amplitude, which we perceive as a louder sound or brighter light.
  • Destructive Interference: Occurs when waves are out of phase, such that the crest of one wave meets the trough of another. This results in the waves canceling each other out, reducing the amplitude and often producing silence or darkness.
Wave interference is a core concept in physics because it explains the behavior and properties of waves in various mediums, from sound and light to water waves. By understanding interference, we can predict how waves will interact, which is crucial for engineering, music production, and even astronomy.
Time Interval for Maxima
The time interval between successive maxima (or beats) in a wave interference pattern is crucial for understanding how often these maximum points, or peaks, occur over time.

To determine this interval, we use the formula for the period \(T\) of the beat frequency:
  • \(T = \frac{1}{f_{beat}}\)
Given two waves with frequencies \(n_1\) and \(n_2\), their beat frequency is \(|n_1 - n_2|\). Hence, the time interval \(T\) between successive maxima is:
  • \(T = \frac{1}{|n_1 - n_2|}\)
This formula tells us how long it takes between each occurrence of maximum reinforcement of the combined waves. This concept is particularly important for tuning musical instruments and in applications like sonar and radar, where precise timing of wave peaks is necessary for accurate measurements.

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Most popular questions from this chapter

When two progressive waves of intensity \(I_{1}\) and \(I_{2}\) but slightly different frequencies superpose, the resultant intensity fluctuates between (a) \(\left(\sqrt{I_{1}}+\sqrt{I_{2}}\right)^{2}\) and \(\left(\sqrt{I_{1}}-\sqrt{I_{2}}\right)^{2}\) (b) \(\left(\sqrt{I_{1}}+\sqrt{I_{2}}\right)\) and \(\left(\sqrt{I_{1}}-\sqrt{I_{2}}\right)\) (c) \(\left(I_{1}+I_{2}\right)\) and \(\sqrt{I_{1}-I_{2}}\) (d) \(\frac{I_{1}}{I_{2}}\) and \(\frac{I_{2}}{I_{1}}\)

In the case of stationary waves all the particles of the medium between two nodes vibrate (a) in phase but with different amplitudes and time periods (b) in phase and with same amplitude and time period (c) in phase with the same time period but different amplitudes (d) with the same time period but in different phases and with different amplitudes

A window whose area is \(2 \mathrm{~m}^{2}\) opens on a street where street noises result in an intensity level at the window of \(60 \mathrm{~dB}\). How much acoustic power enters the window via sound waves and if an acoustic absorber is fitted at the window, how much energy from street noise will it collect in five hours? (a) \(3 \mu \mathrm{W}, 2 \times 10^{-3} \mathrm{~J}\) (b) \(2 \mu \mathrm{W}, 36 \times 10^{-3} \mathrm{~J}\) (c) \(36 \mu \mathrm{W}, 2 \times 10^{-3} \mathrm{~J}\) (d) \(2 \mu \mathrm{W}, 3.6 \times 10^{-3} \mathrm{~J}\)

An organ pipe open at both the ends and another organ pipe closed at one end will resonate with each other, if their lengths are in the ratio of (a) \(1: 1\) (b) \(1: 4\) (c) \(2: 1\) (d) \(1: 2\)

Two waves are represented by: \(y_{1}=4 \sin 404 \pi t\) and \(y_{2}=3 \sin 400 \pi \mathrm{t}\). Then (a) beat frequency is \(4 \mathrm{~Hz}\) and the ratio of maximum to minimum intensity is \(49: 1\) (b) beat frequency is \(2 \mathrm{~Hz}\) and the ratio of maximum to minimum intensity is \(49: 1\) (c) beat frequency is \(2 \mathrm{~Hz}\) and the ratio of maximum to minimum intensity is I : 49 (d) beat frequency is \(4 \mathrm{~Hz}\) and the ratio of maximum to minimum intensity is \(1: 49\)
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