/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Two sinusoidal waves with identi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 40

Two sinusoidal waves with identical wavelengths and amplitudes travel in opposite directions along a string with a speed of \(10 \mathrm{~cm} / \mathrm{s}\). If the time interval between instants when the string is flat is \(0.50 \mathrm{~s}\), what is the wavelength of the waves?

Short Answer

Expert verified
The wavelength of the waves is 10 cm.

Step by step solution

01

Understand the problem

We have two sinusoidal waves traveling in opposite directions along a string, creating a standing wave. The string is flat when the wave crests and troughs overlap, which occurs twice per wavelength.
02

Identify the given values

The speed of the wave (\(v\)) is given as \(10 \, \text{cm/s}\), and the time interval between the string being flat is \(0.50 \, \text{s}\). This is the time for half a wavelength.
03

Analyze the frequency of the wave

The string becomes flat twice every wavelength. Therefore, the time for a complete wavelength to pass is twice the given time interval, so the period \(T\) is \(2 \times 0.50 \, \text{s} = 1.00 \, \text{s}\).
04

Use the wave speed formula

The relationship between speed \(v\), wavelength \(\lambda\), and period \(T\) is given by the formula \(v = \frac{\lambda}{T}\). Rearranging gives us \(\lambda = v \times T\).
05

Calculate the wavelength

Substitute the known values into the formula: \(\lambda = 10 \, \text{cm/s} \times 1.00 \, \text{s} = 10 \, \text{cm}\).

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

standing wave
A standing wave is a pattern created when two identical waves travel in opposite directions in the same medium. It typically occurs on a string or in a column of air. Standing waves are characterized by nodes, which are points that remain stationary, and antinodes, where maximum amplitude occurs.
Standing waves often develop due to wave reflection, such as when waves reflect back upon reaching the end of a string. The waves interfere constructively and destructively at different points, leading to the standing wave pattern.
This pattern is significant in many physical systems, including musical instruments, where standing waves define the notes and harmonics produced. When observing a standing wave on a string, the nodes are the points that appear stationary, while the antinodes are where the maximum vibration occurs.
wavelength calculation
Wavelength is a key property of waves and is the distance between two corresponding points on successive waves, such as crest to crest or trough to trough.
To calculate the wavelength when dealing with standing waves, it's essential to understand the time interval given, which represents half a wavelength. In the problem above, the time interval of 0.50 seconds indicates that the string becomes flat twice each full wave cycle. Therefore, to find the time of a complete wave, the period (T), is doubled:
  • The period, T, is calculated as: \[ T = 2 \times 0.50 \, \text{s} = 1.00 \, \text{s} \]
With the period known, the actual wavelength is derived from the relationship between speed and period. Thus, using these givens, helps in determining the value of the wavelength for the wave system.
wave speed formula
The wave speed formula is a fundamental relationship in wave mechanics that connects wave speed, frequency, and wavelength.
This formula is expressed as:
  • \[ v = \frac{\lambda}{T} \]
Where,
  • v = the speed of the wave (in cm/s),
  • \( \lambda \) = the wavelength (in cm),
  • T = the period of the wave (in seconds).
By rearranging the formula, we solve for wavelength (\( \lambda \)):
  • \[ \lambda = v \times T \]
This relationship shows that the wavelength is directly proportional to both the speed and period of the wave. Utilizing this formula, students can easily calculate the wavelength when given the speed and period, as demonstrated in the original word problem. Understanding this formula allows a greater grasp of wave behaviors and their calculations.

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 speed of electromagnetic waves (which include visible light, radio, and \(x\) rays \()\) in vacuum is \(3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\). (a) Wavelengths of visible light waves range from about \(400 \mathrm{~nm}\) in the violet to about \(700 \mathrm{~nm}\) in the red. What is the range of frequencies of these waves? (b) The range of frequencies for shortwave radio (for example, FM radio and VHF television) is \(1.5\) to \(300 \mathrm{MHz}\). What is the corresponding wavelength range? (c) X-ray wavelengths range from about \(5.0 \mathrm{~nm}\) to about \(1.0 \times 10^{-2} \mathrm{~nm} .\) What is the frequency range for \(x\) rays?

What are (a) the lowest frequency, (b) the second lowest frequency, and (c) the third lowest frequency for standing waves on a wire that is \(10.0 \mathrm{~m}\) long, has a mass of \(100 \mathrm{~g}\), and is stretched under a tension of \(250 \mathrm{~N}\) ?

A sand scorpion can detect the motion of a nearby beetle (its prey) by the waves the motion sends along the sand surface (Fig. \(16-29\) ). The waves are of two types: transverse waves traveling at \(v_{t}=50 \mathrm{~m} / \mathrm{s}\) and longitudinal waves traveling at \(v_{l}=150 \mathrm{~m} / \mathrm{s} .\) If a sudden motion sends out such waves, a scorpion can tell the distance of the beetle from the difference \(\Delta t\) in the arrival times of the waves at its leg nearest the beetle. If \(\Delta t=4.0 \mathrm{~ms}\) what is the beetle's distance?

String \(A\) is stretched between two clamps separated by distance \(L\). String \(B\), with the same linear density and under the same tension as string \(A\), is stretched between two clamps separated by distance \(4 L\). Consider the first eight harmonics of string \(B\). For which of these eight harmonics of \(B\) (if any) does the frequency match the frequency of (a) \(A\) 's first harmonic, (b) \(A\) 's second harmonic, and (c) \(A\) 's third harmonic?

A sinusoidal wave of frequency \(500 \mathrm{~Hz}\) has a speed of \(350 \mathrm{~m} / \mathrm{s}\). (a) How far apart are two points that differ in phase by \(\pi / 3\) rad? (b) What is the phase difference between two displacements at a certain point at times \(1.00 \mathrm{~ms}\) apart?
See all solutions

Recommended explanations on Physics Textbooks

Physics of Motion

Read Explanation

Circular Motion and Gravitation

Read Explanation

Wave Optics

Read Explanation

Classical Mechanics

Read Explanation

Linear Momentum

Read Explanation

Nuclear Physics

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.