Copyright © 2018 John Wiley & Sons, Inc. and primarily advanced by Prof. A. Iskandar.
16-1 Transverse Waves (1 of 11) Link to heading
Learning Objectives
- 16.01 Identify the three main types of waves.
- 16.02 Distinguish between transverse waves and longitudinal waves.
- 16.03 Given a displacement function for a traverse wave, determine amplitude $y_m$, angular wave number $k$, angular frequency $\omega$, phase constant $\phi$, and direction of travel, and calculate the phase $kx \pm \omega t + \phi$ and the displacement at any given time and position.
16-1 Transverse Waves (2 of 11) Link to heading
- 16.04 Given a displacement function for a traverse wave, calculate the time between two given displacements.
- 16.05 Sketch a graph of a transverse wave as a function of position, identifying amplitude $y_m$ wavelength $\lambda$, where the slope is greatest, where it is zero, and where the string elements have positive velocity, negative velocity, and zero velocity.
- 16.06 Given a graph of displacement versus time for a transverse wave, determine amplitude $y_m$ and period $T$.
16-1 Transverse Waves (3 of 11) Link to heading
- 16.07 Describe the effect on a transverse wave of changing phase constant $\phi$.
- 16.08 Apply the relation between the wave speed $v$, the distance traveled by the wave, and the time required for that travel.
- 16.09 Apply the relationships between wave speed $v$, angular frequency $\omega$, angular wave number $k$, wavelength $\lambda$, period $T$, and frequency $f$.
16-1 Transverse Waves (4 of 11) Link to heading
- 16.10 Describe the motion of a string element as a transverse wave moves through its location, and identify when its transverse speed is zero and when it is maximum.
- 16.11 Calculate the transverse velocity $u(t)$ of a string element as a transverse wave moves through its location.
- 16.12 Calculate the transverse acceleration $a(t)$ of a string element as a transverse wave moves through its location.
- 16.13 Given a graph of displacement, transverse velocity, or transverse acceleration, determine the phase constant $\phi$.
16-2 Wave Speed on a Stretched String (1 of 2) Link to heading
Learning Objectives
- 16.14 Calculate the linear density $\mu$ of a uniform string in terms of the total mass and total length.
- 16.15 Apply the relationship between wave speed $ν$, tension $\tau$, and linear density $\mu$.
16-3 Energy and Power of a Wave Traveling along a String (1 of 3) Link to heading
Learning Objective
- 16.16 Calculate the average rate at which energy is transported by a transverse wave.
16-4 The Wave Equation (1 of 4) Link to heading
Learning Objective
- 16.17 For the equation giving a string-element displacement as a function of position $x$ and time $t$, apply the relationship between the second derivative with respect to $x$ and the second derivative with respect to $t$.
16-5 Interference of Waves (1 of 6) Link to heading
Learning Objective
- 16.18 Apply the principle of superposition to show that two overlapping waves add algebraically to give a resultant (or net) wave.
- 16.19 For two transverse waves with the same amplitude and wavelength and that travel together, find the displacement equation for the resultant wave and calculate the amplitude in terms of the individual wave amplitude and the phase difference.
16-5 Interference of Waves (2 of 6) Link to heading
- 16.20 Describe how the phase difference between two transverse waves (with the same amplitude and wavelength) can result in fully constructive interference, fully destructive interference, and intermediate interference.
- 16.21 With the phase difference between two interfering waves expressed in terms of wavelengths, quickly determine the type of interference the waves have.
16-6 Phasors (1 of 3) Link to heading
Learning Objective
- 16.22 Using sketches, explain how a phasor can represent the oscillations of a string element as a wave travels through its location.
- 16.23 Sketch a phasor diagram for two overlapping waves traveling together on a string, indicating their amplitudes and phase difference on the sketch.
- 16.24 By using phasors, find the resultant wave of two transverse waves traveling together along a string, calculating the amplitude and phase and writing out the displacement equation, and then displaying all three phasors in a phasor diagram that shows the amplitudes, the leading or lagging, and the relative phases.
16-7 Standing Waves and Resonance (1 of 4) Link to heading
Learning Objective
- 16.25 For two overlapping waves (same amplitude and wavelength) that are traveling in opposite directions, sketch snapshots of the resultant wave, indicating nodes and antinodes.
- 16.26 For two overlapping waves (same amplitude and wavelength) that are traveling in opposite directions, find the displacement equation for the resultant wave and calculate the amplitude in terms of the individual wave amplitude.
- 16.27 Describe the SHM of a string element at an antinode of a standing wave.
16-7 Standing Waves and Resonance (2 of 4) Link to heading
- 16.28 For a string element at an antinode of a standing wave, write equations for the displacement, transverse velocity, and transverse acceleration as functions of time.
- 16.29 Distinguish between “hard” and “soft” reflections of string waves at a boundary.
- 16.30 Describe resonance on a string tied taut between two supports, and sketch the first several standing wave patterns, indicating nodes and antinodes.
- 16.31 In terms of string length, determine the wavelengths required for the first several harmonics on a string under tension.
- 16.32 For any given harmonic, apply the relationship between frequency, wave speed, and string length.
Summary (1 of 4) Link to heading
- Waves
- Transverse Waves
- Longitudinal Waves
- Wave Speed
- $\frac{\rm Angular \ velocity}{\rm Angular wavenumber}$ $$\tag{16-13} v = \frac{\omega}{k} = \frac{\lambda}{T} = \lambda f. $$
- Sinusoidal Waves
- Wave moving in positive direction (vector) $$\tag{16-2} y(x,t) = y _m \sin (kx - \omega t) $$
- Traveling Waves
- A functional form for traveling waves $$\tag{16-17} y(x,t) = h(kx \pm \omega t) $$
- Power
- Average Powers is given by $$\tag{16-33} P_{\rm avg} = \frac12 \mu v \omega^2 y_m^2 $$
- Standing Waves
- The interference of two identical sinusoidal waves moving in opposite directions produces standing waves. $$\tag{16-60} y’(x,t) = [2 y_m \sin kx] \cos \omega t $$
- Interference of Waves
- Two sinusoidal waves on the same string exhibit interference $$\tag{16-51} y’(x,t) = [2 y_m \cos \tfrac12 \phi] \sin (kx - \omega t + \tfrac12 \phi). $$
- Resonance
- For a stretched string of length L with fixed ends, the resonant frequencies are $$\tag{16-66} f = \frac{v}{\lambda} = n \frac{v}{2L}, \ \ \ \ {\rm for} \ n = 1, 2, 3, \dots. $$
Copyright Link to heading
Copyright © 2018 John Wiley & Sons, Inc.
All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
C. SECTION 117 COMPUTER PROGRAM EXEMPTIONS Link to heading
Section 117 of the Copyright Act of 1976 was enacted in the Computer Software Copyright Amendments of 1980 in response to the recommendations of the National Commission on New Technological Uses of Copyrighted Works’ (CONTU). Section 117 permits the owner of a copy of a computer program to make an additional copy of the program for purely archival purposes if all archival copies are destroyed in the event that continued possession of the computer program should cease to be rightful, or where the making of such a copy is an essential step in the utilization of the computer program in conjunction with a machine and that it is used in no other manner.