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Chapter 12: Problem 10

A wave is represented by the equation \(\mathrm{y}=0.5 \sin (10 \mathrm{t}+\mathrm{x}) \mathrm{m}\). It is a travelling wave propagating along the \(x\) direction with velocity. [Roorkee 1995] (a) \(10 \mathrm{~m} / \mathrm{s}\) (b) \(20 \mathrm{~m} / \mathrm{s}\) (c) \(5 \mathrm{~m} / \mathrm{s}\) (d) None of these

Short Answer

Expert verified
The velocity of the wave is \(10 m/s\), so option (a) is correct.

Step by step solution

01

Identification of angular frequency and wave number

In the equation given, \(y = 0.5 \sin (10t + x)\), we can identify that \( \omega \) (angular frequency) is 10 and \(k\) (wave number) which is accompanies \(x\) is 1.
02

Calculate the wave velocity

Using the formula, \(v = \omega / k\), we substitute \( \omega \) = 10 and \(k\) = 1 into the equation. This gives us \(v = 10 / 1 = 10 m/s\).

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Key Concepts

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

Angular Frequency
Angular frequency is an essential concept in the study of waves, and it's symbolized by \( \omega \). It reflects how many oscillations occur in a unit of time, typically measured in radians per second. In the given wave equation \( y = 0.5 \sin(10t + x) \), the angular frequency \( \omega \) can be directly identified by the coefficient of the time variable \( t \), which is 10.

This number can be thought of as how fast the wave oscillates per second. If we imagine a swinging pendulum, the angular frequency would tell us how many swings it makes back and forth each second, scaled by the angle in radians.

Here's why angular frequency matters in context:
  • It determines the rate of change of the wave's phase over time.
  • Higher angular frequency implies more cycles per unit time, meaning the wave oscillates faster.
  • The angular frequency can be linked to other properties of the wave, like wave number and velocity.
Wave Number
The wave number, denoted by \( k \), describes the spatial characteristics of a wave. It's essentially the number of wave cycles per unit distance, typically measured in radians per meter. In the context of the provided wave equation \( y = 0.5 \sin(10t + x) \), the wave number \( k \) is extracted from the coefficient of the position variable \( x \), which is 1.

Think of the wave number as a measure of how many oscillations fit within a particular distance in space. It's critical because it relates to how compressed or stretched a wave appears.

For better understanding, consider why wave number is important:
  • The wave number is directly related to the wavelength \( \lambda \), since \( k = \frac{2\pi}{\lambda} \).
  • Higher wave numbers mean that more complete cycles fit within a designated space, indicating shorter wavelengths.
  • It helps to determine the wave patterns that develop in various mediums, which has implications in fields like acoustics and optics.
Wave Velocity
Wave velocity, symbolized by \( v \), indicates the speed and direction in which the wave propagates through space. It is the speed at which the wavefronts (crest, troughs) move along the medium. For the given wave \( y = 0.5 \sin(10t + x) \), we calculate wave velocity using the formula:

\[ v = \frac{\omega}{k} \]

Substituting the angular frequency (\( \omega = 10 \)) and wave number (\( k = 1 \)), we find that \( v = \frac{10}{1} = 10 \) m/s.

This calculation shows that the wave moves at 10 meters per second in the positive x-direction.

Understanding wave velocity involves recognizing different factors:
  • Wave velocity is affected by the medium through which the wave travels. Different materials and conditions can alter the speed.
  • It combines both angular frequency and wave number, illustrating the harmonious relationship between time-based and spatial characteristics.
  • This parameter is crucial in many practical applications, including predicting the behavior of sound, light, and water waves.

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Most popular questions from this chapter

The stationary wave produced on a string is represented by the equation \(y=5 \cos \left(\frac{\pi x}{3}\right) \sin (40 \pi t)\) where \(x\) and \(y\) are in \(\mathrm{cm}\) and \(t\) is in seconds. The distance between consecutive nodes is (a) \(5 \mathrm{~cm}\) (b) \(\pi \mathrm{cm}\) (c) \(3 \mathrm{~cm}\) (d) \(40 \mathrm{~cm}\)

A motor car blowing a horn of frequency 124 vibration/sec moves with a velocity \(72 \mathrm{~km} / \mathrm{hr}\) towards a tall wall. The frequency of the reflected sound heard by the driver will be (velocity of sound in air is \(330 \mathrm{~m} / \mathrm{s}\) ) [MP PET 1997] (a) 109 vibration \(/ \mathrm{sec}\) (b) 132 vibration \(/ \mathrm{sec}\) (c) 140 vibration \(/ \mathrm{sec}\) (d) 248 vibration \(/ \mathrm{sec}\)

On sounding tuning fork \(A\) with another tuning fork B of frequency \(384 \mathrm{~Hz}, 6\) beats are produced per second. After loading the prongs of \(A\) with wax and then sounding it again with \(B, 4\) Beats are produced per second what is the frequency of the tuning fork \(A\). (a) \(388 \mathrm{~Hz}\) (b) \(8 \mathrm{o} \mathrm{Hz}\) (c) \(378 \mathrm{~Hz}\) (d) \(390 \mathrm{~Hz}\)

In a wave motion \(y=a \sin (k x-\omega t), y\) can represents (a) Electric Field (b) magnetic field (c) Displacement (d) Pressure

The driver of car travelling with a speed 30 meter/sec. towards a hill sounds a horn of frequency \(600 \mathrm{~Hz}\). If the velocity of sound in air is \(330 \mathrm{~m} / \mathrm{s}\) the frequency of reflected sound as heard by the driver is [MP PMT \(\mathbf{1 9}\) (a) \(720 \mathrm{~Hz}\) (b) \(555.5 \mathrm{~Hz}\) (c) \(550 \mathrm{~Hz}\) (d) \(500 \mathrm{~Hz}\)
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