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Chapter 12: Problem 6

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

Short Answer

Expert verified
(c) Displacement

Step by step solution

01

Understanding wave motion

In a wave motion, the wave equation \(y=a \sin (k x-\omega t)\) represents a sinusoidal wave. Here, 'y' typically represents the disturbance or variation from a fixed point caused by the wave at any given point of time.
02

Analysing the options

Let's analyze each option individually: (a) Electric Field: In electromagnetic waves, the electric field varies sinusoidally, but the equation for this variation is not typically presented as \(y=a \sin (k x-\omega t)\). (b) Magnetic Field: Similar to the electric field, this refers to electromagnetic waves, but again, its equation of variation is not as given. (c) Displacement: In mechanical waves such as sound waves or waves on a string, 'y' often refers to the displacement of particles from their equilibrium positions. The given equation fits this case. (d) Pressure: In a sound wave, pressure variations follow a sinusoidal pattern, but are often not represented with the given equation.
03

Conclusion

Through our analysis, it is seen that 'y' in the wave equation \(y=a \sin (k x-\omega t)\) closely matches the Displacement in a wave motion, where 鈥榶鈥� can be the displacement of particles from their equilibrium positions when the wave passes through.

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Key Concepts

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

Wave Equation
The wave equation is a fundamental concept used to describe the behavior of waves in various mediums. It is typically expressed as:\[ y = a \sin(kx - \omega t) \]where:
  • \(y\) is the wave function representing the displacement at a point \(x\) and time \(t\).
  • \(a\) is the amplitude, which measures the maximum strength or height of the wave.
  • \(k\) is the wave number, defining the number of wave cycles per unit distance.
  • \(\omega\) is the angular frequency, indicating how many oscillations occur per unit time.
This equation describes sinusoidal waves, which are prevalent in various fields, such as acoustics and optics. The wave equation provides a mathematical framework to analyze how waves propagate through space over time.
Sinusoidal Wave
A sinusoidal wave is a type of smooth periodic oscillation that is mathematically described by a sine or cosine function. This kind of wave is perfect for modeling physical wave situations because of its well-understood and predictable properties.
  • Its repeating pattern is caused by the nature of the sine function, which is continuous and symmetric.
  • The amplitude determines the wave's height, contributing to the energy it carries.
  • Sinusoidal waves are common in nature, seen in sound waves, light waves, and even in electrical signals.
Understanding sinusoidal waves is crucial for comprehending more complex wave behavior and for applications ranging from music to engineering design.
Mechanical Waves
Mechanical waves are disturbances that require a medium to travel through, such as solids, liquids, or gases. Unlike electromagnetic waves, they cannot propagate through a vacuum. Common types of mechanical waves include:
  • Sound waves: compressional waves traveling through air or other media.
  • Water waves: disturbances moving across the surface of a water body.
  • Seismic waves: waves generated by geological activities like earthquakes.
Mechanical waves transport energy from one location to another and often cause the particles of the medium to oscillate and then return to their equilibrium position. For precise analysis, wave equations are used to describe how mechanical waves behave under various conditions.
Displacement in Waves
Displacement in waves relates to how much a particle of the medium moves from its rest position as a wave passes through. In the context of the wave equation:\[ y = a \sin(kx - \omega t) \]\(y\) represents the displacement of particles. Here's what this means for wave motion:
  • Displacement is typically maximum at the wave crest or trough and zero as it passes through the equilibrium point.
  • In mechanical waves like those on a string, displacement can be viewed as the vertical movement of the string segments.
  • Understanding displacement helps in visualizing the wave shape and predicting how the wave behaves when traveling through different media.
This concept is critical in analyzing energy transfer, signal transmission, and understanding the real-world applications of wave dynamics.

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Most popular questions from this chapter

When two sound waves with a phase difference of \(\pi / 2\) and each having amplitude \(A\) and frequency \(\omega\) are superimposed on each other, then the maximum amplitude and frequency of resultant wave is \(\quad\) [MP PMT 1989] (a) \(\frac{A}{\sqrt{2}} ; \omega / 2\) (b) \(\frac{A}{\sqrt{2}} ; \omega\) (c) \(\sqrt{2} A ; \frac{\omega}{2}\) (d) \(\sqrt{2} A ; \omega\)

A source of sound of frequency \(500 \mathrm{~Hz}\) is moving towards an observer with velocity \(30 \mathrm{~m} / \mathrm{s}\). The speed of the sound is \(330 \mathrm{~m} / \mathrm{s}\). The frequency heard by the observer will be [MP PET 2000] (a) \(550 \mathrm{~Hz}\) (b) \(458.3 \mathrm{~Hz}\) (c) \(530 \mathrm{~Hz}\) (d) \(545.5 \mathrm{~Hz}\)

A tuning fork \(F_{1}\) has a frequency of \(256 \mathrm{~Hz}\) and it is observed to produce 6 beats/second with another tuning fork \(F_{2}\). When \(F_{2}\) is loaded with wax. It still produces 6 beats/second with \(F_{1}\). The frequency of \(F_{2}\) before loading was (a) \(253 \mathrm{~Hz}\) (b) \(262 \mathrm{~Hz}\) (c) \(250 \mathrm{~Hz}\) (d) \(259 \mathrm{~Hz}\)

A wave represented by the given equation \(y=A \sin \left(10 \pi x+15 \pi t+\frac{\pi}{3}\right)\) where \(x\) is in meter and \(t\) is in second. The expression represents (a) A wave travelling in the positive \(x\)-direction with a velocity of \(1.5 \mathrm{~m} / \mathrm{sec}\) (b) A wave travelling in the negative \(x\)-direction with a velocity of \(1.5 \mathrm{~m} / \mathrm{sec}\) (c) A wave travelling in the negative \(x\)-direction with a wavelength of o.2 \(m\) (d) A wave travelling in the positive \(x\)-direction with a wavelength of \(0.2 \mathrm{~m}\)

The equation of a transverse wave travelling on a rope is given by \(y=10 \sin \pi(0.01 x-2.00 t)\) where \(y\) and \(x\) are in \(\mathrm{cm}\) and \(t\) in seconds. The maximum transverse speed of a particle in the rope is about [MP PET 1999] (a) \(63 \mathrm{~cm} / \mathrm{sec}\) (b) \(75 \mathrm{~cm} / \mathrm{s}\) (c) \(100 \mathrm{~cm} / \mathrm{sec}\) (d) \(121 \mathrm{~cm} / \mathrm{sec}\)
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