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Chapter 10: Problem 1487

Which of the following changes at an antinode in a stationary wave? (A) Density only (B) Pressure only (C) Both pressure and density (D) Neither pressure nor density

Short Answer

Expert verified
At an antinode in a stationary wave, particles oscillate with maximum amplitude, causing them to become compressed or rarefied at different moments of their oscillation. This results in changes in both density and pressure. Therefore, the correct answer is (C) Both pressure and density.

Step by step solution

01

Understand stationary waves and antinodes

A stationary wave, also known as a standing wave, is formed when two waves of the same frequency, amplitude, and wavelength traveling in opposite directions interfere with each other. An antinode is the position in a stationary wave where the amplitude of oscillation is maximum. That means the particles at an antinode have the maximum displacement.
02

Analyze the properties of antinodes

The density of a medium is defined as its mass per unit volume, whereas pressure is defined as a force exerted per unit area. At an antinode, particles oscillate with maximum amplitude, which means that their displacement from their equilibrium position is maximum. Because of this displacement, the particles can become compressed or rarefied at different moments of their oscillation.
03

Determine the change in density and pressure at an antinode

When the particles of the medium are compressed, they are closer together, and therefore, the density is higher, and so is the pressure. In contrast, when the particles are rarefied, they are farther apart, leading to lower density and pressure. Since the particles oscillate between being compressed and rarefied, both density and pressure change at an antinode in a stationary wave.
04

Choose the correct option

Based on the analysis, we can conclude that both density and pressure change at an antinode in a stationary wave. Therefore, the correct answer is: (C) Both pressure and density

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Key Concepts

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

Antinode
An antinode in a stationary wave is a key point where maximum movement occurs, showing the highest amplitude. This is where the oscillating particles reach their furthest points from their usual resting positions. Unlike nodes, which have no movement, antinodes are vibrant and dynamic.

At an antinode, the energy of the wave is at its peak, causing significant displacement of particles. This displacement results in frequent changes in various properties of the wave, such as pressure and density. Understanding these changes helps grasp the dynamic nature of waves fully.
Density Change
Density change in a stationary wave is most noticeable at the antinode. Since density measures how compact the particles of a medium are, it tends to fluctuate when these particles move. During the oscillation at an antinode, particles are alternately compressed and rarefied.

When particles are bunched together during compression, density increases because more mass occupies a smaller volume. Conversely, when particles spread out during rarefaction, density decreases. These variations are direct reflections of the oscillatory motion happening within the wave.
Pressure Change
Just like with density, pressure changes radically at the antinode of a stationary wave. Pressure is the force exerted by particles per unit area, and as particles move, the force they exert also changes.

When particles compress, they exert more force on each other, increasing pressure. Conversely, during rarefaction, the force decreases as particles move apart, leading to lower pressure. This interplay of compression and rarefaction at the antinode highlights how pressure is tightly linked to the wave鈥檚 motion and its energy distribution.
Interference of Waves
The interference of waves is the principle that allows stationary waves or standing waves to form. When two waves with the same frequency and amplitude travel through a medium in opposite directions, they overlap. This overlapping is called interference.

Interference can be constructive or destructive. Constructive interference occurs when the wave crests (the highest points) and troughs (the lowest points) of the two waves align, creating points of maximum amplitude or antinodes. On the other hand, destructive interference leads to nodes, where the waves cancel each other out, resulting in no movement. This beautiful dance of alignment and cancellation produces the characteristic pattern of a stationary wave.

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Most popular questions from this chapter

A simple pendulum is executing S.H.M. around point \(\mathrm{O}\) between the end points \(B\) and \(C\) with a periodic time of \(6 \mathrm{~s}\). If the distance between \(\mathrm{B}\) and \(\mathrm{C}\) is \(20 \mathrm{~cm}\) then in what time will the bob move from \(C\) to \(D\) ? Point \(D\) is at the mid-point of \(C\) and \(\mathrm{O}\). (A) \(1 \mathrm{~s}\) (B) \(2 \mathrm{~s}\) (C) \(3 \mathrm{~s}\) (D) \(4 \mathrm{~s}\)

The bob of a simple pendulum having length ' \(\ell\) ' is displaced from its equilibrium position by an angle of \(\theta\) and released. If the velocity of the bob, while passing through its equilibrium position is \(\mathrm{v}\), then \(\mathrm{v}=\ldots \ldots \ldots\) (A) \(\sqrt{\\{2 g \ell(1-\cos \theta)\\}}\) (B) \(\sqrt{\\{2 g \ell(1+\sin \theta)\\}}\) (C) \(\sqrt{\\{2 g \ell(1-\sin \theta)\\}}\) (D) \(\sqrt{\\{2 g \ell(1+\cos \theta)\\}}\)

Two waves are represented by \(\mathrm{y}_{1}=\mathrm{A} \sin \omega \mathrm{t}\) and \(\mathrm{y}_{2}=\mathrm{A}\) cos \(\omega \mathrm{t}\). The phase of the first wave, \(\mathrm{w}\). r. t. to the second wave is (A) more by radian (B) less by \(\pi\) radian (C) more by \(\pi / 2\) (D) less by \(\pi / 2\)

If the equation for a particle performing S.H.M. is given by \(\mathrm{y}=\sin 2 \mathrm{t}+\sqrt{3} \cos 2 \mathrm{t}\), its periodic time will be \(\ldots \ldots .\) s. (A) 21 (B) \(\pi\) (C) \(2 \pi\) (D) \(4 \pi\).

In a longitudinal wave, pressure variation and displacement variation are (A) In phase (B) \(90^{\circ}\) out of phase (C) \(45^{\circ}\) out of phase (D) \(180^{\circ}\) out of phase
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