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Chapter 17: Problem 5

Show That Show that $$ \begin{array}{ll} y(x, t)=Y \sin k(x-v t), & y(x, t)=Y \sin 2 \pi\left(\frac{x}{\lambda}-f t\right) \\ y(x, t)=Y \sin \omega\left(\frac{x}{v}-t\right), & y(x, t)=Y \sin 2 \pi\left(\frac{x}{\lambda}-\frac{t}{T}\right) \end{array} $$ are all equivalent to \(y(x, t)=Y \sin (k x-\omega t)\).

Short Answer

Expert verified
All expressions transform into \( y(x, t) = Y \sin (kx - \omega t) \) using given relationships.

Step by step solution

01

Understand the given expressions

Recognize each given waveform expression and write down what each term represents. Let鈥檚 compare them with the target expression: \(y(x, t) = Y \sin (kx - \omega t)\).
02

Compare the first expression

For \(y(x, t)=Y \sin k(x-vt)\), recognize that: \(k = 2\pi/\lambda \) and \(v = \omega /k \). Substituting these values gives \( y(x, t) = Y \sin (kx - \omega t) \).
03

Compare the second expression

For \(y(x, t) = Y \sin 2 \pi \left( \frac{x}{\lambda} - f t \right)\), recognize that: \(k = 2\pi/\lambda \) and \(\omega = 2\pi f \). Substituting these into \( y(x, t) = Y\sin (kx - \omega t) \), it matches.
04

Compare the third expression

For \(y(x, t) = Y \sin \omega \left( \frac{x}{v} - t \right)\), recognize \(k = \omega / v\). Substituting these gives \( y(x, t) = Y\sin (kx - \omega t) \).
05

Compare the fourth expression

For \(y(x, t) = Y \sin 2 \pi \left( \frac{x}{\lambda} - \frac{t}{T} \right)\), recognize that: \(k = 2\pi/\lambda \) and \(\omega = 2\pi/T \). Substituting these gives \( y(x, t) = Y \sin (kx - \omega t) \).

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Key Concepts

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

Wave Function
A wave function is a mathematical representation of a physical wave. It helps to describe how a wave propagates through space and time. The general form of a wave function is typically written as \(y(x, t) = Y \text{sin}(kx - \frac{\tau}{\text{t}})\), where different parameters define the specific characteristics of the wave. The wave function encapsulates information about the wave's amplitude (Y), its wavenumber (k), angular frequency (\tau), and phase velocity (v). It is a crucial tool in physics to model the behavior of waves in various contexts, such as sound waves, light waves, and water waves.
Sine Wave
A sine wave is one of the simplest and most fundamental types of waveforms. It represents a smooth, periodic oscillation. Its mathematical form in the context of waves is typically written as \( y(x, t) = Y \text{sin}(kx - \tau t)\). Key characteristics of a sine wave include:
  • Amplitude (Y): The height of the wave, which represents the maximum displacement from its rest position.
  • Wavelength (\tau): The distance over which the wave repeats.
  • Frequency: The number of oscillations the wave undergoes in a unit of time.
Sine waves are prevalent in various natural phenomena, including sound and electromagnetic waves.
Angular Frequency
Angular frequency refers to how quickly the wave oscillates in radians per second. It is denoted by the Greek letter omega (饾湐). The mathematical relationship is given by \( \tau = 2\text{pi} f \), where 'f' represents the frequency in hertz (cycles per second). Angular frequency provides insight into the temporal aspect of the wave's propagation. It tells us how rapidly the wave cycles through its phases. For instance, a higher angular frequency means the wave oscillates more quickly. You can encapsulate \( \tau \) with \text{sin}(kx - 饾湐t) to understand how it affects the wave.
Wavenumber
The wavenumber is a measure of spatial frequency, representing the number of wave cycles per unit distance. It is denoted by the letter 'k' and is calculated as \( k = 2\text{pi /}\tau\). Here, '饾渾' represents the wavelength, the distance over which the wave repeats. The wavenumber offers a spatial perspective on the wave, indicating how many wavelengths fit into a unit of distance. It's critical for understanding and calculating phase shifts in wave mechanics. When plugged into the wave function \( y(x, t) = Y \text{sin}(kx - \tau t)\), 'k' transforms our idea from temporal to spatial dimensions.
Phase Velocity
Phase velocity refers to the rate at which the phase of the wave propagates in space. It is defined by the equation \( v = 饾湐 / k\), where 'k' is the wavenumber and '饾湐' is the angular frequency. Phase velocity helps to comprehend how quickly the wavefronts (lines of constant phase) move. For example, in the equation \( y(x, t) = Y \text{sin}(k(x - vt))\), 'v' represents the phase velocity. It shows that different sine waves can move through the medium at varying speeds, depending on their frequency and wavelength. Phase velocity is a key parameter in wave studies, affecting how waves interact, reflect, and transmit through different mediums.

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Most popular questions from this chapter

Displacement of Particles A sinusoidal wave is traveling on a string with speed \(40 \mathrm{~cm} / \mathrm{s}\). The displacement of the particles of the string at \(x=10 \mathrm{~cm}\) is found to vary with time according to the equation \(y(x, t)=(5.0 \mathrm{~cm}) \sin [1.0 \mathrm{rad} / \mathrm{cm}-(4.0 \mathrm{rad} / \mathrm{s}) t] .\) The linear density of the string is \(4.0 \mathrm{~g} / \mathrm{cm} .\) What are the (a) frequency and (b) wavelength of the wave? (c) Write the general equation giving the transverse displacement of the particles of the string as a function of position and time. (d) Calculate the tension in the string.

The Fastest Wave What is the fastest transverse wave that can be sent along a steel wire? For safety reasons, the maximum tensile stress to which steel wires should be subjected is \(7.0 \times 10^{8} \mathrm{~N} / \mathrm{m}^{2}\). The density of steel is \(7800 \mathrm{~kg} / \mathrm{m}^{3}\). Show that your answer does not depend on the diameter of the wire.

Three-Loop Standing Wave A string \(3.0 \mathrm{~m}\) long is oscillating as a three-loop standing wave with an amplitude of \(1.0 \mathrm{~cm}\). The wave speed is \(100 \mathrm{~m} / \mathrm{s}\). (a) What is the frequency? (b) Write equations for two waves that, when combined, will result in this standing wave.

Average Power A string along which waves can travel is \(2.70 \mathrm{~m}\) long and has a mass of \(260 \mathrm{~g}\). The tension in the string is \(36.0 \mathrm{~N}\). What must be the frequency of traveling waves of amplitude \(7.70\) \(\mathrm{mm}\) for the average power to be \(85.0 \mathrm{~W} ?\)

Linear Density The linear density of a string is \(1.6 \times 10^{-4} \mathrm{~kg} / \mathrm{m}\). A transverse wave on the string is described by the equation $$ y(x, t)=(0.021 \mathrm{~m}) \sin [(2.0 \mathrm{rad} / \mathrm{m}) x+(30 \mathrm{rad} / \mathrm{s}) t] $$ What is (a) the wave speed and (b) the tension in the string?
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