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Chapter 10: Problem 99

Which onc of the following do not represent a travelling harmonic wave? (a) \(A(a x-b t)\) (b) \(A \tan (\omega t-k x)\) (c) \(\Lambda \cos ^{2}(v t-x)\) (d) \(A \cos k x \sin \omega t\)

Short Answer

Expert verified
Both options (a) and (b) do not represent traveling harmonic waves.

Step by step solution

01

Understanding the Structure of a Traveling Wave

A traveling wave is typically expressed in the form of a function such as \(y(x, t) = A heta(kx \pm \omega t)\), where \(A\) is the amplitude, \(k\) is the wave number, and \(\omega\) is the angular frequency. The argument of the function represents a phase, which changes linearly with both space \(x\) and time \(t\).
02

Analyzing Option (a)

Option (a) is given as \(A(a x-b t)\). This form lacks a periodic function such as sine or cosine. Instead, it is simply a linear combination of \(x\) and \(t\). Therefore, this does not exhibit the properties of a harmonic wave.
03

Analyzing Option (b)

Option (b) is \(A \tan(\omega t - k x)\). Although it includes a function with terms \(x\) and \(t\) that are linear, \(\tan\) is not typically used in traveling waves due to its vertical asymptotes, which make it non-periodic over all \(x\) and \(t\).
04

Analyzing Option (c)

Option (c) is \(\Lambda \cos^2(v t - x)\). By using the identity \(\cos^2\theta = \frac{1 + \cos(2\theta)}{2}\), this can be rewritten as a combination of constant and a \(\cos\) term with twice the argument, making it a form of a wave. Thus, it can represent a traveling wave.
05

Analyzing Option (d)

Option (d) is \(A \cos kx \sin \omega t\). Using the product-to-sum identity, this expression can be rewritten as a difference of two wave functions: \(\frac{A}{2}[\sin(kx + \omega t) - \sin(kx - \omega t)]\). Hence, it represents a combination of traveling waves.
06

Conclusion

Analyzing each option, it is clear that option (a) does not contain a sinusoidal function, and option (b), while having linear \(x\) and \(t\) terms, uses a non-periodic \(\tan\) function. Thus, options (c) and (d) are both valid representations of traveling waves.

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Key Concepts

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

Harmonic Waves
Harmonic waves are specific types of waves that repeat themselves at regular intervals, both in time and space. These waves exhibit a periodic motion and are often described using sinusoidal functions, such as sine or cosine functions. This periodic nature gives harmonic waves their characteristic oscillating pattern.

To better understand harmonic waves, it's important to note:
  • They have a consistent and repeating structure.
  • They are described by formulas involving sinusoidal functions like \(B\sin(kx - \omega t)\cos(kx + \omega t)\cos^2(kx + \omega t)\frac{1}{2}(1 + \cos(2\theta))\right\).
  • Harmonic waves are used extensively in physics to analyze vibrations, sound waves, and light waves.
Wave Function
The wave function is a crucial concept when studying waves. It represents the displacement of a point at any location \(x\) and time \(t\) in the medium through which the wave is traveling. Typically, a wave function for a harmonic wave is expressed as \(y(x, t) = A \mathbb{X}(kx \pm \omega t)\), where \(\mathbb{X}\) represents a trigonometric function, such as sine or cosine.

Key aspects of wave functions include:
  • The amplitude \(A\), which indicates the maximum displacement of the wave.
  • The phase, expressed as \(kx \pm \omega t\), which determines the position and time characteristics of the wave.
  • Wave functions can describe how waves move through space and time, providing insights into wave behaviors.
Wave Equation
The wave equation is a mathematical representation that describes the behavior of a wave. It provides information about how a wave propagates through a medium. A hallmark equation used in this context is the one-dimensional wave equation: \[ \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2} \], where \(v\) is the speed of the wave.

Important components of the wave equation include:
  • The partial differential terms that indicate the dependence on both time and space.
  • The speed \(v\), which denotes how fast the wave travels through the medium.
  • This equation is fundamental in understanding wave phenomena across different physical contexts.
Wave Number
The wave number \(k\) plays a significant role in characterizing harmonic waves. Defined as \( k = \frac{2\pi}{\lambda} \), where \(\lambda\) is the wavelength, it signifies how many wave cycles exist over a unit length.

Some highlights about wave number include:
  • It's directly related to the wavelength鈥攕horter wavelengths result in higher wave numbers.
  • Wave number is used in wave equations and functions to determine the spatial frequency of a wave.
  • This concept is essential for analyzing patterns and structures of waves in various fields such as optics and quantum mechanics.
Angular Frequency
Angular frequency \(\omega\) is a fundamental concept in the study of harmonic waves. It describes how quickly the wave oscillates in time and is measured in radians per second, defined as \( \omega = 2\pi f \), where \(f\) is the frequency.

Key points about angular frequency include:
  • It provides insight into the temporal oscillations of the wave.
  • Higher angular frequencies correspond to faster oscillating waves.
  • Angular frequency is used alongside wave number to give a full description of a wave's motion through space and time.

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Most popular questions from this chapter

Equation of a wave travelling in a medium is: \(y=a \sin (b t-c x) .\) Which of the following are correct? (a) Ratio of the displacement ampliude, with which the particles of the medium oscillate, to the wavelength is equal to ac/2\pi. (b) Ratio of the velocily oscillation amplitude of medium particles to the wave propagation velocity is equal to \(a c\). (c) Oscillation amplitude of relative deformation of the medium is direcly proportional to velocity oscillation amplitude of medium particles. (d) None of the above

Temperature dependence of speed of sound could be expressed as, \(v=k \sqrt{T}\), where \(k\) is some positive constant, In a medium sound is travelling along \(x\)-axis. At \(x=0\), temperature is \(T_{1}\) and at \(x=l\), temperature is \(T_{2}\) and the temperature varies linearly with \(x\). Time taken by sound to travel from \(x=0\) to \(x=l\) is (a) \(\frac{2 l}{k\left(\sqrt{T_{1}}+\sqrt{T_{2}}\right)}\) (b) \(\frac{l}{k\left(\sqrt{T_{1}}+\sqrt{T_{2}}\right)}\) (c) \(\frac{l}{2 k\left(\sqrt{T_{1}}+\sqrt{T_{2}}\right)}\) (d) \(\frac{l}{2 k\left(\sqrt{T_{1}}+\sqrt{T_{2}}\right)^{2}}\)

Statement-1 : Oscillatory motions are necessarily periodic motions, but all periodic motions are not oscillatory. Statement-2 : Simple pendulum is an example of oscillatory motion.

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 $$ \omega=\sqrt{\frac{k}{m}}=\sqrt{\frac{20}{0.2}}=10 \mathrm{rad} / \mathrm{s} $$

\(\Lambda\) wave represented by \(y=2 \cos (4 x-\pi t)\) is supcrposed with another wave to form a stationary wave such that the point \(x=0\) is a node. The equation of other wave is (a) \(2 \sin (4 x+\pi t)\) (b) \(-2 \cos (4 x-\pi t)\) (c) \(-2 \cos (4 x+\pi t)\) (d) \(-2 \sin (4 x-\pi t)\)
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