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Chapter 14: Problem 34

Mark correct option or options (a) Any function \(y(x, t)=f(\omega t+k x)\) repre- sents a progressive wave (b) The stationary wave on a string under tension fixed at end does not have well defined nodes (c) The phenomenon of beats is not observed in the case of visible light waves (d) All of the above

Short Answer

Expert verified
Correct options are (a) and (c).

Step by step solution

01

Analyze Option (a)

A progressive wave is generally given by the expression \( y(x, t) = A \sin(\omega t + kx + \phi) \) or \( y(x,t) = A \cos(\omega t + kx + \phi) \), where \( \omega \) is the angular frequency and \( k \) is the wave number. This matches the form \( y(x, t) = f(\omega t + kx) \). Hence, option (a) is correct as it fits the description of a progressive wave.
02

Analyze Option (b)

A stationary wave can be described by the sum of two progressive waves traveling in opposite directions. For nodes to form, which are points of zero amplitude, there must be a fixed pattern. In the case where a string is fixed at both ends, nodes are formed at regular intervals, thus option (b) is incorrect.
03

Analyze Option (c)

Beats occur due to the interference of two waves of slightly different frequencies. Although beats are commonly studied with sound waves, they can occur with any type of waves, including light. However, the beat phenomenon in visible light is not easily observed due to high frequencies and the wave nature of light. Therefore, option (c) is correct.
04

Consider Option (d)

Option (d) states 'All of the above'. Since only options (a) and (c) are correct based on our analysis, option (d) must be incorrect, as not all options from (a) to (c) are correct.

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Key Concepts

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

stationary wave
A stationary wave, also known as a standing wave, is formed when two progressive waves of the same frequency and amplitude travel in opposite directions and combine. This process results in a pattern that appears to be standing still. Unlike progressive waves that move through the medium, stationary waves do not transfer energy in the direction of wave propagation.

Stationary waves are essential in many applications, such as musical instruments and architectural acoustics. When a string is fixed at both ends, such as in a guitar or piano, stationary waves are created. The
  • wave nodes, defined as points of zero amplitude, appear at fixed positions.
  • Antinodes, conversely, are points of maximum amplitude.

The regular pattern of nodes and antinodes is due to the constructive and destructive interference of the waves. The interference causes stationary points, or nodes, along the medium where the waves cancel each other out. In contrast, the points between nodes experience constant oscillation and are referred to as antinodes.
beats phenomenon
The phenomenon of beats is a fascinating aspect of wave behavior, occurring due to the interference of two waves with slightly different frequencies. When these waves overlap, they create a new wave pattern that varies in amplitude over time.

The fluctuating amplitude results in alternating loud and soft sounds, heard as beats. This periodic change in sound intensity is the key distinctive feature of beats.
  • Beats can be observed in sound waves, making it a common demonstration in physics and music.
  • For visible light, the occurrence of beats is theoretically possible but not easily observed.
Due to the high frequencies involved in visible light, the rapid fluctuation between the light and dark areas is not typically perceptible to the human eye. Nevertheless, understanding this phenomenon in terms of waves enhances our comprehension of wave interactions beyond everyday experiences.
nodes and antinodes formation
Nodes and antinodes are fundamental concepts in the study of waves, particularly in stationary waves. They help us visualize and understand how waves behave when reflected or confined.

Nodes are points along a wave where there is no movement; the wave amplitude is zero. These points occur at fixed intervals and are created by the destructive interference of two overlapping waves.

Antinodes, on the other hand, are points of maximum amplitude. At these positions, constructive interference leads to the maximum displacement of the wave.

  • In a vibrating string fixed at both ends, nodes are found at the fixed points, ensuring the string does not move there.
  • Antinodes occur between these nodes, where the string vibrates most.

This formation allows us to analyze wave behavior and frequency in various materials, including strings, air columns, and surfaces. The predictable pattern of nodes and antinodes in stationary waves provides the foundation for numerous applications, from musical instruments to advanced communication technology.

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Most popular questions from this chapter

A sound wave of pressure amplitude 14 pascal propagates through the air medium. The normal pressure of air is \(1.0 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2} .\) The difference between maximum and minimum pressure in the medium is : (a) \(5 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\) (b) \(10 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\) (c) \(10 \mathrm{~N} / \mathrm{m}^{2}\) (d) none of these

A source of sound \(S\) having frequency of generated sound \(300 \mathrm{~Hz}\) is moving in a circle of radius \(2 \mathrm{~m}\) with angular speed \(10 / \pi\) rps. A detector \(D\) in the plane of the circle is at a distance of \(30 \mathrm{~m}\) from the centre. The speed of sound in air is \(300 \mathrm{~m} / \mathrm{s}\), then : (a) the maximum frequency detected at \(D\) will be \(340 \mathrm{~Hz}\) (b) the minimum frequency detected at \(D\) will be \(280 \mathrm{~Hz}\) (c) the average frequency of listening the frequency 300 \(\mathrm{Hz}\) is \(\frac{20}{\pi}\) per second (d) the source reaches \(A\) when maximum frequency is detected at \(D\)

A source at rest sends waves of constant wavelength. \(\mathrm{A}\) wall moves towards the source with a velocity \(33 \mathrm{~m} / \mathrm{s}\). The velocity of sound in the medium is \(330 \mathrm{~m} / \mathrm{s}\). What is the percentage change in wavelength of sound after reflection from the wall? (a) \(0.1 \%\) (b) \(2 \%\) (c) \(9.1 \%\) (d) \(1 \%\)

A disc of radius \(R\) is rotating uniformly with angular frequency \(\omega .\) A source of sound is fixed to the rim of the disc. The ratio of maximum and minimum frequencies heard by stationary observer, far away from the disc and in the plane of the disc is : (Given: \(v=\) speed of sound) (a) \(\left(\frac{v}{v-R \omega}\right)\) (b) \(\left(\frac{v}{v+R \omega}\right)\) (c) \(\left(\frac{v-R \omega}{v+R \omega}\right)\) (d) \(\left(\frac{v+R \omega}{v-R \omega}\right)\)

When temperature of air is \(20^{\circ} \mathrm{C}\), a tuning fork sounded over the open end of an air column produces 4 beats per second, the tuning fork given a lower note. If the frequency of tuning fork is \(34 \mathrm{~Hz}\), then how many beats will be produced by the tuning fork if temperature falls to \(5^{\circ} \mathrm{C} ?\) (a) 2 beat \(/ \mathrm{sec}\) (b) 4 beat \(/ \mathrm{sec}\) (c) 1 beat \(/ \mathrm{sec}\) (d) 3 beat \(/ \mathrm{sec}\)
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