Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 15: Problem 16

Consider a monochromatic wave on a string, with amplitude \(A\) and wavelength \(\lambda\), traveling in one direction. Find the relationship between the maximum speed of any portion of string, \(v_{\max },\) and the wave speed, \(v\)

Short Answer

Expert verified
Answer: The maximum speed of any portion of the string, \(v_{\max}\), is related to the wave speed, \(v\), through the equation: $$ v_{\max} = 2\pi A f $$ where \(A\) is the amplitude of the wave, and \(f\) is the frequency. The frequency can be expressed in terms of the wave speed and wavelength through the equation \(v = f\lambda\).

Step by step solution

01

Understanding the properties of a monochromatic wave on a string

A monochromatic wave on a string is a wave with a single frequency. The wave is characterized by its amplitude, \(A\), and wavelength, \(\lambda\). The amplitude is the maximum displacement of the string from its equilibrium position, and the wavelength is the distance between two consecutive points with the same phase (e.g. two consecutive peaks or troughs) in the wave.
02

Writing down the wave equation

The wave equation that describes a monochromatic wave on a string traveling in one direction can be written as: $$ y(x,t)=A\sin\left(\frac{2\pi}{\lambda}(x-vt)\right) $$ where \(y(x,t)\) represents the displacement of the string as a function of position, \(x\), and time, \(t\), and \(v\) is the wave speed.
03

Calculating the velocity of a point on the wave

To find the velocity of a point on the wave, we need to determine the rate of change of the displacement with respect to time. We can do this by taking the partial derivative of the wave equation with respect to time: $$ v_y(x,t) = \frac{\partial y(x,t)}{\partial t} $$ Applying the chain rule and differentiating the wave equation with respect to time: $$ v_y(x,t) = A \frac{2\pi}{\lambda} (-v) \cos\left(\frac{2\pi}{\lambda}(x-vt)\right) $$
04

Finding the maximum value of the velocity

To find the maximum speed of any portion of the string, we need to find the maximum value of \(v_y(x,t)\). Since the maximum value of the cosine function is \(1\), the maximum value of \(v_y(x,t)\) is obtained when \(\cos\left(\frac{2\pi}{\lambda}(x-vt)\right)=1\). Hence, $$ v_{\max} = A \frac{2\pi}{\lambda}(-v) $$
05

Expressing the maximum speed in terms of the wave speed

To find the relationship between the maximum speed, \(v_{\max}\), and the wave speed, \(v\), we can rearrange the equation we derived in the previous step: $$ v_{\max} = 2\pi A \frac{-v}{\lambda} $$ Since the frequency, \(f\), is related to the wave speed and wavelength by \(v=f\lambda\), we can rewrite the above equation in terms of frequency: $$ v_{\max} = 2\pi A f $$ This equation shows the relationship between the maximum speed of any portion of the string, \(v_{\max}\), and the wave speed, \(v\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The tension in a 2.7 -m-long, 1.0 -cm-diameter steel cable \(\left(\rho=7800 \mathrm{~kg} / \mathrm{m}^{3}\right)\) is \(840 \mathrm{~N}\). What is the fundamental frequency of vibration of the cable?

A sinusoidal wave traveling in the positive \(x\) -direction has a wavelength of \(12 \mathrm{~cm},\) a frequency of \(10.0 \mathrm{~Hz},\) and an amplitude of \(10.0 \mathrm{~cm}\). The part of the wave that is at the origin at \(t=0\) has a vertical displacement of \(5.00 \mathrm{~cm} .\) For this wave, determine the a) wave number, d) speed, b) period, e) phase angle, and c) angular frequency, f) equation of motion.

The displacement from equilibrium caused by a wave on a string is given by \(y(x, t)=(-0.00200 \mathrm{~m}) \sin \left[\left(40.0 \mathrm{~m}^{-1}\right) x-\right.\) \(\left.\left(800 . \mathrm{s}^{-1}\right) t\right] .\) For this wave, what are the (a) amplitude, (b) number of waves in \(1.00 \mathrm{~m},\) (c) number of complete cycles in \(1.00 \mathrm{~s},\) (d) wavelength, and (e) speed?

Suppose that the tension is doubled for a string on which a standing wave is propagated. How will the velocity of the standing wave change? a) It will double. c) It will be multiplied by \(\sqrt{2}\). b) It will quadruple. d) It will be multiplied by \(\frac{1}{2}\).

A string is made to oscillate, and a standing wave with three antinodes is created. If the tension in the string is increased by a factor of 4 a) the number of antinodes increases. b) the number of antinodes remains the same. c) the number of antinodes decreases. d) the number of antinodes will equal the number of nodes.
See all solutions

Recommended explanations on Physics Textbooks

Scientific Method Physics

Read Explanation

Electricity

Read Explanation

Translational Dynamics

Read Explanation

Torque and Rotational Motion

Read Explanation

Measurements

Read Explanation

Work Energy and Power

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free