PreClass Notes: Chapter 14, Sections

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PreClass Notes: Chapter 14, Sections 14.1-14.4 From Essential University Physics 3 rd Edition by Richard Wolfson, Middlebury College 2016 by Pearson Education, Inc.

Outline A wave involves a disturbance that moves or propagates through space. The disturbance carries energy, but not matter.

1 PreClass Notes: Chapter 14, Sections From Essential University Physics 3 rd Edition by Richard Wolfson, Middlebury College 2016 by Pearson Education, Inc. Narration and extra little notes by Jason Harlow, University of Toronto This video is meant for University of Toronto students taking PHY131. Outline A wave involves a disturbance that moves or propagates through space. The disturbance carries energy, but not matter. Air doesn t move from your mouth to a listeners ear, but sound energy does. R.Wolfson [Image of waves in a field of grass from ] Wave Properties The Wave Equation Waves on a string Wave Power and Intensity Sound Waves 1

transported any distance. [image from https://webspace.utexas.edu/cokerwr/www/index.

2 What is a Wave? A wave is a traveling disturbance that transports energy but not matter. Mechanical waves are disturbances of a material medium. The medium moves briefly as the wave goes by, but the medium itself isn't transported any distance. [image from by Daniel A. Russell ] Electromagnetic waves, including light, do not require a medium. Transverse Waves In a transverse wave, the disturbance is perpendicular to the wave motion. 2

Properties of Continuous Waves Wavelength λ is the distance over which a wave repeats in space.

3 Longitudinal Waves In a longitudinal wave, the disturbance is parallel to the wave motion. Sound is an longitudinal wave. Properties of Continuous Waves Wavelength λ is the distance over which a wave repeats in space. Period T is the time for a complete oscillation of the wave at a fixed position. Frequency f is the number of wave cycles per unit time: f = 1/T Amplitude A is the maximum value of the wave disturbance. 3

4 Wave Speed Wave speed is the rate at which the wave propagates. Wave speed, wavelength, period, and frequency are related: Simple Harmonic Waves A simple harmonic wave is described by a sinusoidal function of space and time: y x,t ( ) = Acos( kx ±wt) y measures the wave disturbance at position x and time t. k = 2π/λ is the wave number, a measure of the rate at which the wave varies in space. ω = 2π/T is the angular frequency, a measure of the rate at which the wave varies in time. The ± is written so we can describe a wave going in the +x direction ( sign) or the x direction (+ sign). The wave speed is: v = λf = ω k 4

5 Got It? [1 of 5] These two waves have the same speed. Which has the greater amplitude? A B Got It? [2 of 5] These two waves have the same speed. Which has the greater wavelength? A B 5

6 Got It? [3 of 5] These two waves have the same speed. Which has the greater period? A B Got It? [4 of 5] These two waves have the same speed. Which has the greater wave number? A B 6

7 Got It? [5 of 5] These two waves have the same speed. Which has the greater frequency? A B The Wave Equation Many different types of media can support the propagation of waves. Analysis of disturbances in different media often result in the following equation (which is known as the one-dimensional wave equation): 2 y x = 1 2 y 2 v 2 t 2 This equation relates the space and time derivatives of the disturbed quantity, y. Note that v is the wave speed, where: v = l T = l f It can be shown that any function of the form satisfies the wave equation. y = f (x ± vt) 7

8 Waves on Strings On strings, fibers, long springs, cables, wires, etc., tension provides the restoring force that helps transverse waves propagate. Newton's second law gives 2Fq = mv2 R 2q Rmv2 = R = 2qmv 2 where F is the tension and μ is the mass per unit length. The speed of such waves is v = F m Wave Power The power carried by a wave is proportional to the wave speed and to the square of the wave amplitude. For waves on a string, the average power is: P = 1 2 mw 2 A 2 v Other types of waves have similar equations for average power, always dependent on the square of the wave amplitude, A 2. 8

P A = P 4pr 2 Sound Sound waves are longitudinal mechanical waves that propagate through gases, liquids, and solids.

9 Wave Intensity Wave intensity is the power crossing a unit perpendicular area. In a plane wave, the intensity remains constant. A spherical wave spreads in three dimensions, so its intensity drops as the inverse square of the distance from its source: I = P A = P 4pr 2 Sound Sound waves are longitudinal mechanical waves that propagate through gases, liquids, and solids. Sound waves in air involve small changes in air pressure and density, associated with back-and-forth motion of the air as the wave passes. 9

10 The Speed of Sound The speed of sound in a gas depends on the background pressure P, density ρ, and a factor γ that is determined by the number of atoms that form a typical gas molecule: v = g P r The speed of sound is determined by the air and is not dependent upon the amplitude, frequency, or wavelength of the sound itself. In dry air at atmospheric pressure and room temperature, the speed of sound is v = 343 m/s. Human Hearing and the Decibel The human ear responds to a broad range of sound intensities and frequencies The audible range extends from about 20 Hz to 20 khz in frequency and over 12 orders of magnitude in intensity The sound intensity level β is measured in decibels: ( ) b = 10log I I 0 The decibel is a logarithmic unit based on a comparison to the nominal threshold of hearing: I 0 = W / m 2 10

11 Human Hearing and the Decibel 11

Physics Fall Mechanics, Thermodynamics, Waves, Fluids. Lecture 37: Wave motion II. Slide 37-1

Physics Fall Mechanics, Thermodynamics, Waves, Fluids. Lecture 37: Wave motion II. Slide 37-1 Physics 1501 Fall 2008 Mechanics, Thermodynamics, Waves, Fluids Lecture 37: Wave motion II Slide 37-1 Recap: properties of waves Wavelength λ is the distance over which a wave repeats in space. Period