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Chapter 11: Problem 92

A sine wave has an amplitude \(A\) and wavelength \(\lambda\) The ratio of particle velocity and the wave velocity is \((2 \pi A=\lambda)\) (a) \(\leq 1\) (b) \(=1\) (c) \(\geq 1\) (d) None of these

Short Answer

Expert verified
The ratio of particle velocity to wave velocity is 1 when \(2\pi A = \lambda\).

Step by step solution

01

Understanding the Given Data

We are provided with a sine wave characterized by an amplitude, \(A\), and a wavelength, \(\lambda\). We need to evaluate the ratio of the particle velocity to wave velocity specifically when \(2\pi A = \lambda\).
02

Defining Particle and Wave Velocities

The particle velocity \(v_p\) is the derivative of the displacement with respect to time. For a sine wave given by \(y = A \sin(kx - \omega t)\), where \(k = \frac{2\pi}{\lambda}\) and \(\omega\) is the angular frequency. The wave velocity \(v_w\) is given by \(v_w = \frac{\omega}{k} = \frac{\omega \lambda}{2\pi}\).
03

Calculating Particle Velocity

The particle velocity \(v_p\) is the time derivative of the displacement function. Differentiate \(y = A \sin(kx - \omega t)\) with respect to \(t\), which yields \( v_p = -A \omega \cos(kx - \omega t) \). The maximum particle velocity occurs when \( \cos(kx - \omega t) = \pm 1 \), giving \( |v_p|_{max} = A \omega \).
04

Substituting Given Relation

From the provided condition \(2\pi A = \lambda\), substitute this into the expression for wave velocity: \( v_w = \frac{\omega \lambda}{2\pi} = \frac{\omega \cdot 2\pi A}{2\pi} = \omega A \). Thus, \(v_w = A \omega\).
05

Evaluating the Ratio

Substitute \(v_p = A \omega \) and \(v_w = A \omega \) into the ratio: \( \text{Ratio} = \frac{v_p}{v_w} = \frac{A \omega}{A \omega} = 1 \).
06

Choosing the Correct Answer

Seeing that the ratio simplified to 1, the correct answer choice is (b) \(=1\).

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Key Concepts

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

Particle Velocity
When discussing wave motion, it's important to understand particle velocity. This is a measure of how quickly a particular point, or particle, on the wave moves as the wave passes through it. For a sine wave, the equation of motion can be written as
  • \(y = A \sin(kx - \omega t)\)
The particle velocity, denoted as \(v_p\), is found by differentiating the displacement \(y\) concerning time \(t\). This gives:
  • \(v_p = -A \omega \cos(kx - \omega t)\)
The maximum particle velocity occurs when \(\cos(kx - \omega t) = \pm 1\), resulting in \(|v_p|_{max} = A \omega\). This shows how quickly the wave鈥檚 form vibrates along with the medium due to the wave's passage.
Wave Velocity
Wave velocity, often represented as \(v_w\), indicates how quickly the wave propagates through the medium. For a sine wave, the formula connects the wavelength \(\lambda\) and angular frequency \(\omega\):
  • \(v_w = \frac{\omega}{k} = \frac{\omega \lambda}{2\pi}\)
Here, \(k = \frac{2\pi}{\lambda}\) is the wave number, representing the number of wavelengths per unit distance. Understanding this is crucial as it differentiates the speed of the physical wave motion from the speed at which individual particles in the medium move (particle velocity). The wave velocity tells us how fast the entire wave front moves through space.
Sine Wave
The sine wave is a fundamental concept in understanding wave motion. It refers to a smooth periodic oscillation that is often used to model waves. This is represented mathematically as:
  • \(y = A \sin(kx - \omega t)\)
Where \(A\) is the amplitude, \(kx\) involves the spatial variation, and \(\omega t\) involves the temporal variation. The sine function describes how the wave increases and decreases as it propagates, creating peaks and troughs that characterize this type of wave. Understanding sine waves helps in visualizing oscillatory systems such as sound waves, light waves, and even waves in water.
Wavelength
Wavelength, denoted as \(\lambda\), is an essential property of waves. It is the distance between successive crests, troughs, or identical points in consecutive cycles of a wave. In a sine wave, the wavelength determines how 'stretched' the wave appears:
  • A higher wavelength means a broader wave, while a smaller wavelength means a more compressed wave.
This spatial period is inversely related to the wave number \(k\), with \(k = \frac{2\pi}{\lambda}\). The wavelength is crucial in determining the wave speed when combined with the frequency of the wave, following the simple relation of
  • \(\text{Wave Speed} = \text{Frequency} \times \text{Wavelength}\)
Amplitude
Amplitude is a measure of the wave's energy and determines the height of the wave. For a sine wave, amplitude \(A\) indicates the maximum extent of a wave鈥檚 oscillation from its resting position. It measures how 'loud' a sound wave is or how 'bright' a light wave appears:
  • Greater amplitude indicates more energy in the wave.
  • It is symmetrically distributed on both sides of the central axis of the wave.
In the formula of wave motion, \(y = A \sin(kx - \omega t)\), amplitude sets the scale of the wave's peaks and troughs. Therefore, amplitude plays a vital role in determining how impactful the effects of the wave will be on the medium through which it is traveling.

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Most popular questions from this chapter

A wave equation is \(y=10^{-4} \sin (60 t+2 x)\) where, \(x\) and \(y\) are in metres and \(t\) is in second. Which of the following statements is correct? (a) The wave travels with a velocity of \(300 \mathrm{~m} / \mathrm{s}\) in the negative direction of the \(x\) -axis. (b) Its wavelength is \(\pi\) metre. (c) Its frequency is \(50 \pi\) hertz. (d) All of these

A simple harmonic progressive wave is represented by the equation \(y=8 \sin 2 \pi(0.1 x-2 t)\) where, \(x\) and \(y\) are in \(\mathrm{cm}\) and \(t\) is in seconds. At any instant the phase difference between two particles separated by \(2.0 \mathrm{~cm}\) in the \(X\) -direction is (a) \(18^{\circ}\) (b) \(54^{\circ}\) (c) \(36^{\circ}\) (d) \(72^{\circ}\)

A wave equation which gives the displacement along \(Y\) -direction is given \(y=10^{4} \sin (60 t+2 x)\) where, \(x\) and \(y\) are in metre and \(t\) in sec. Among the following choose the correct statement (a) It represents a wave propagating along positive \(x\) -axis with a velocity of \(30 \mathrm{~m} / \mathrm{s}\). (b) It represents a wave propagating along negative \(x\) -axis with a velocity of \(120 \mathrm{~m} / \mathrm{s}\). (c) It represents a wave propagating along negative \(x\) -axis with a velocity of \(30 \mathrm{~m} / \mathrm{s}\). (d) It represents a wave propagating along negative \(x\) -axis with a velocity of \(10^{4} \mathrm{~m} / \mathrm{s}\).

A particle is subjected simultaneously to two SHM's, one along the \(x\) -axis and the other along the \(y\) -axis. The two vibrations are in phase and have unequal amplitudes. The particle will execute (a) straight line motion (b) circular motion (c) elliptic motion (d) parabolic motion

In the arrangement, spring constant \(k\) has value \(2 \mathrm{~N}\) \(\mathrm{m}^{-1}\), mass \(M=3 \mathrm{~kg}\) and mass \(m=1 \mathrm{~kg}\). Mass \(M\) is in contact with a smooth surface. The coefficient of friction between two blocks is \(0.1\). The time period of SHM executed by the system is (a) \(\pi \sqrt{6}\) (b) \(\pi \sqrt{2}\) (c) \(2 \sqrt{2} \pi\) (d) \(2 \pi\)
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