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Chapter 10: Problem 98

Which one of the following represent a travelling wave? (a) \(y-A \sqrt{(x-v)}\) (b) \(y=A \cos (a x+b c)\) (c) \(y=A \log (x-v t)\) (d) \(y=f\left(x^{2}-y^{2}\right)\)

Short Answer

Expert verified
Option (c) \(y = A \log (x - vt)\) is a traveling wave.

Step by step solution

01

Understand the Form of a Traveling Wave

A traveling wave is typically represented by a function of the form \(y = f(x \, \pm \, vt)\) or \(y = f(x - vt)\) or \(y = f(x + vt)\), where \(v\) is the wave's velocity and \(t\) is time. This form indicates a wave moving through space.
02

Analyze Option (a)

Option (a) is given as \(y = A \sqrt{(x-v)}\). For a wave function, we expect it to depend on \(x\) and \(t\) combined in the form \(x \pm vt\). Here, the function depends only on \(x-v\), and there is no time \(t\) component, so it cannot represent a traveling wave.
03

Analyze Option (b)

Option (b) is given as \(y = A \cos (a x + b c)\). This function indicates a cosine wave in space but not dependent on time \(t\), and \(b\) and \(c\) do not clearly represent time. A traveling wave must involve both \(x\) and \(t\), so this cannot be a traveling wave.
04

Analyze Option (c)

Option (c) is \(y = A \log (x - vt)\). This function depends on the combination \(x - vt\), which fits the requirement of a traveling wave form \(y = f(x - vt)\). Hence, it represents a traveling wave.
05

Analyze Option (d)

Option (d) is \(y = f\left(x^{2}-y^{2}\right)\). This function depends on \(x^{2} - y^{2}\), which does not align with the form \(x \pm vt\). Thus, it cannot be a traveling wave.

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Key Concepts

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

Wave Function
The wave function is a fundamental tool used to describe waves in mathematics and physics. It tells us the displacement of a point on the wave as a function of space and time. In the simplest form, a wave function is expressed as:
  • One-dimensional, like \(x\): \([\text{shape of wave}(t, x)]\)
  • Depends on both space and time variables, implying motion across regions.
  • Shows how the wave evolves over time.
For example, a wave traveling along a string can be represented as \(y(x, t) = A \sin(kx - \omega t)\), where \(A\) is the amplitude, \(k\) is the wave number, and \(\omega\) the angular frequency. Understanding the wave function helps us predict the behavior of waves over time and in different spatial areas.
Wave Velocity
Wave velocity refers to the speed at which any point on a wave, like the crest, propagates through a medium. It indicates how fast the wave travels and is crucial in understanding wave dynamics. The wave velocity \(v\) can be driven by different factors:
  • The medium through which the wave travels.
  • The type of wave (e.g., transverse or longitudinal).
In mathematical terms, the relationship can be succinctly described as \(v = f\lambda\), where \(f\) is the frequency and \(\lambda\) is the wavelength of the wave. Recognizing the wave velocity's significance is key to solving various physical problems related to wave propagation, like using it to distinguish between traveling and standing waves.
Wave Equation
The wave equation is a fundamental part of physics that models how waves propagate through different mediums. It's a partial differential equation, widely used to describe mechanical waves, electromagnetic waves, and quantum mechanical waves. The standard form of the wave equation is:
  • One-dimensional: \(\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}\)
  • Relates the displacement \(y\), speed \(v\), and time \(t\).
Solutions to the wave equation describe the shape and motion of waves, supporting the development of mathematical models that can predict how waves travel in various contexts. It provides a unified way to understand the myriad of wave-based phenomena found in nature.
Cosine Wave
A cosine wave is a specific type of wave, often used for its mathematical properties. It's similar to a sine wave, differing only in a phase shift. This wave can be represented by the equation \(y = A \cos(kx - \omega t)\) where:
  • \(A\) is the amplitude, showing the wave鈥檚 height.
  • \(kx - \omega t\) represents the phase, demonstrating movement over time and space.
  • It's commonly used in fields like electronics, signal processing, and physics.
The cosine wave oscillates between its peak and valley, making it a perfect model for describing periodic phenomena like sound waves. It鈥檚 particularly useful because its symmetrical properties simplify calculations, making it a favorite in theoretical physics and engineering disciplines.

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Most popular questions from this chapter

In the state of rotation of the system as a whole there are two forces which act on the sphere: (i) elastic and (ii) centripetal. The equation of motion of the sphere is \(\frac{m d^{2} x}{d t^{2}}=-k x+m \omega^{2} x ;\) (here \(x\) sunds for displacement) or \(\frac{d^{2} x}{d t^{2}}=-\left(\frac{k}{m}-\omega^{2}\right) x\) i.e., the sphere will execute S.H.M. with $$ \begin{aligned} \omega_{n} &=\sqrt{\frac{k}{m}-\omega^{2}} \\ \therefore \quad T &=\frac{2 \pi}{\omega_{o}}=\frac{2 \pi}{\sqrt{k / m-\omega^{2}}} \\ &=\frac{2 \times 3.14}{\sqrt{\frac{20}{0.2}-(4.4 \times 4.4)}} \approx 0.7 \mathrm{~s} \end{aligned} $$ The oscillation will stop when \(T=\infty\), This is posible when $$ \omega=\sqrt{\frac{k}{m}}=\sqrt{\frac{20}{0.2}}=10 \mathrm{rad} / \mathrm{s} $$

Two pendulum have time period \(T\) and \(\frac{37}{4}\). They start \(\mathrm{SHM}\) at the same time from two extreme postion at \(t=0 .\) \Lambdafler how much time they will be in same phasc? (a) \(\frac{T}{2}\) (b) \(\frac{3 T}{2}\) (c) \(\frac{5 T}{2}\) (d) \(\frac{5 T}{4}\)

Statement-1 : Sound wave cannot be polarised. Statement-2 : Polarisation is shown by transversc wavc only.

When a string is vibrating in standing wave pattern the power transmitted across an antinode, compared to power transmitted across node is (a) More (b) Less (c) Same (zero) (d) Same (nonzero)

Two sound sources of same frequency produce sound intensities \(I_{0}\) and \(4 I_{0}\) at a point \(P\) when used separately. Now, they are used together so that the sound waves from them reach \(P\) with a phase difference \(\phi\). Determine the resultant intensity at \(P\) for (a) \(\phi=0\) (b) \(\phi=2 \pi / 3\) (c) \(\phi=\pi\)
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