/*! 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 42 The wave function of a standing ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 15: Problem 42

The wave function of a standing wave is \(y(x, t) = 4.44 \, \mathrm{mm} \, \mathrm{sin}[(32.5 \, \mathrm{rad/m})x] \mathrm{sin}[(754 \, \mathrm{rad/s})t]\). For the two traveling waves that make up this standing wave, find the (a) amplitude; (b) wavelength; (c) frequency; (d) wave speed; (e) wave functions. (f) From the information given, can you determine which harmonic this is? Explain.

Short Answer

Expert verified
The amplitude is 2.22 mm; wavelength is 0.193 m; frequency is 120 Hz; wave speed is 23.2 m/s; wave functions are given. Harmonic can't be determined without medium length.

Step by step solution

01

Understanding the Standing Wave Equation

The given wave function is \(y(x, t) = 4.44 \, \text{mm} \, \sin[(32.5 \, \text{rad/m})x] \sin[(754 \, \text{rad/s})t]\). This is a standing wave, which can be expressed as a sum of two traveling waves moving in opposite directions. By using the trigonometric identity: \( \sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)] \), the given wave can be decomposed into two components.
02

Finding the Amplitude of Traveling Waves

The amplitude of the full standing wave is 4.44 mm. For the two traveling waves, the amplitude is half of the standing wave amplitude since it combines two waves with the same amplitude: \( A = \frac{4.44}{2} = 2.22 \, \text{mm}\).
03

Determining the Wavelength

The wavenumber \( k \) in the equation \( k = 32.5 \, \text{rad/m} \) can be used to find the wavelength with \( \lambda = \frac{2\pi}{k} \). Substituting \( k \), we get \( \lambda = \frac{2\pi}{32.5} \approx 0.193 \, \text{m} \).
04

Calculating the Frequency

The angular frequency \( \omega = 754 \, \text{rad/s} \) relates to the frequency by \( f = \frac{\omega}{2\pi} \). Calculating this gives \( f = \frac{754}{2\pi} \approx 120 \, \text{Hz} \).
05

Finding the Wave Speed

The wave speed \( v \) is found using the formula \( v = f \cdot \lambda \). Substitute the known values: \( v = 120 \, \text{Hz} \times 0.193 \, \text{m} \approx 23.2 \, \text{m/s} \).
06

Writing the Wave Functions

The wave functions for the two traveling waves are \( y_1 = 2.22 \, \text{mm} \, \sin[(32.5 \, \text{rad/m})x - (754 \, \text{rad/s})t] \) and \( y_2 = 2.22 \, \text{mm} \, \sin[(32.5 \, \text{rad/m})x + (754 \, \text{rad/s})t] \). These represent the forward and backward traveling components.
07

Determining the Harmonic

The wavelength \( \lambda = 0.193 \, \text{m} \) can suggest the harmonic if we know the length \( L \) of the medium (string, tube, etc.) containing the wave. If \( L \) were provided, \( n \lambda/2 \) such that \( n \) is an integer would help determine the harmonic. Without \( L \), the harmonic cannot be specified.

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.

Amplitude
Amplitude is a crucial attribute of a wave that defines its maximum displacement from the rest position. For standing waves, the amplitude is typically noted at the antinodes, the points where the wave is at its highest energy. In the given standing wave equation, the amplitude is seen to be 4.44 mm. However, this value represents the combined amplitude of both traveling waves that form the standing wave.
  • To find the amplitude of each individual traveling wave, we divide the standing wave's amplitude by two.
  • This division occurs because the standing wave results from two identical waves moving in opposite directions, each contributing equally to the resultant standing wave amplitude.
  • Thus, each traveling wave has an amplitude of 2.22 mm.
Wavelength
Wavelength helps determine the size of the wave along the medium. It is the distance between consecutive points like crests or troughs. The wavelength defines the spatial period of the wave, dictating how far a wave travels before repeating its cycle.
  • For a wave described by the equation \(2\pi k\), the wavenumber "k" is instrumental in deriving the wavelength \(\lambda\).
  • Through the relation \(\lambda = \frac{2\pi}{k}\), we find that the given standing wave has a wavenumber of 32.5 rad/m.
  • Plugging in this value, we calculate the wavelength to be approximately 0.193 meters.
Frequency
Frequency refers to the number of oscillations that occur in one second. It determines how often particles of the medium vibrate as the wave passes through. In wave functions, the angular frequency \(\omega\) is pivotal in computing the frequency of oscillations.
  • Given \(\omega = 754 \, \text{rad/s}\), the conversion from angular frequency to frequency \(\(f\)\) uses \(f = \frac{\omega}{2\pi}\).
  • Upon calculating, the frequency of each traveling wave forming the standing wave emerges as approximately 120 Hz, meaning 120 cycles per second.
  • This frequency defines how swiftly or slowly the wave pattern repeats over time.
Wave Speed
Wave speed indicates the rate at which energy or information is transmitted through the medium. It is the velocity at which a wave propagates and is influenced by both its frequency and wavelength.
  • The fundamental formula \(v = f \lambda\) is used to calculate wave speed.
  • Using known values, \(v = 120 \, \text{Hz} \times 0.193 \, \text{m}\), we deduce the wave speed as approximately 23.2 m/s.
  • This speed reveals how quickly waves vibrate and spread across the medium.
Wave Functions
Wave functions are equations that describe the motion and properties of waves. In the case of standing waves, two traveling waves superimpose to create a wave pattern that appears stationary. This superposition results from waves moving in opposite directions and, when perfectly aligned, leads to a standing wave.
  • Trigonometric identities like \(\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]\) help express a standing wave in terms of these traveling waves.
  • The provided wave functions \(y_1 = 2.22 \, \text{mm} \, \sin[(32.5 \, \text{rad/m})x - (754 \, \text{rad/s})t]\) and \(y_2 = 2.22 \, \text{mm} \, \sin[(32.5 \, \text{rad/m})x + (754 \, \text{rad/s})t]\) denote these components.
  • This description encapsulates the forward and backward movement that forms the stationary appearance of a standing wave.

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

You are investigating the report of a UFO landing in an isolated portion of New Mexico, and you encounter a strange object that is radiating sound waves uniformly in all directions. Assume that the sound comes from a point source and that you can ignore reflections. You are slowly walking toward the source. When you are 7.5 m from it, you measure its intensity to be 0.11 W/m\(^2\). An intensity of 1.0 W/m\(^2\) is often used as the "threshold of pain." How much closer to the source can you move before the sound intensity reaches this threshold?

A musician tunes the C-string of her instrument to a fundamental frequency of 65.4 Hz. The vibrating portion of the string is 0.600 m long and has a mass of 14.4 g. (a) With what tension must the musician stretch it? (b) What percent increase in tension is needed to increase the frequency from 65.4 Hz to 73.4 Hz, corresponding to a rise in pitch from C to D?

A string with both ends held fixed is vibrating in its third harmonic. The waves have a speed of 192 m/s and a frequency of 240 Hz. The amplitude of the standing wave at an antinode is 0.400 cm. (a) Calculate the amplitude at points on the string a distance of (i) 40.0 cm; (ii) 20.0 cm; and (iii) 10.0 cm from the left end of the string. (b) At each point in part (a), how much time does it take the string to go from its largest upward displacement to its largest downward displacement? (c) Calculate the maximum transverse velocity and the maximum transverse acceleration of the string at each of the points in part (a).

A light wire is tightly stretched with tension F. Transverse traveling waves of amplitude \(A\) and wavelength \(\lambda_1\) carry average power \(P_{av,1} = 0.400\) W. If the wavelength of the waves is doubled, so \(\lambda_2 = 2\lambda_1\), while the tension \(F\) and amplitude \(A\) are not altered, what then is the average power \(P_{av,2}\) carried by the waves?

A piano tuner stretches a steel piano wire with a tension of 800 N. The steel wire is 0.400 m long and has a mass of 3.00 g. (a) What is the frequency of its fundamental mode of vibration? (b) What is the number of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to 10,000 Hz?
See all solutions

Recommended explanations on Physics Textbooks

Magnetism and Electromagnetic Induction

Read Explanation

Oscillations

Read Explanation

Medical Physics

Read Explanation

Measurements

Read Explanation

Fluids

Read Explanation

Energy 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.