jw Fundamentals of Physics 1 Chapter 14 Waves - I 1.Waves & Particles 2.Types of Waves 3.Transverse & Longitudinal Waves 4.Wavelength & Frequency 5.Speed of a Traveling Wave 6.Wave Speed on a Stretched String 7.Energy & Power in a Traveling String Wave 8.The Wave Equation 9.The Principle of Superposition for Waves 10.Interference of Waves 11.Phasors 12.Standing Waves 13.Standing Waves & Resonance

jw Fundamentals of Physics 2 Waves & Particles Particles - a material object moves from one place to another. Waves - information and energy move from one point to another, but no material object makes that journey. –Mechanical waves –Newton’s Laws rule! –Requires a material medium e.g. water, sound, seismic, etc. –Electromagnetic waves –Maxwell’s Equations & 3.0 x 10 8 m/s –No material medium required –Matter waves –Quantum Mechanics - ~ m –Particles have a wave length - De Broglie (1924)

jw Fundamentals of Physics 3 A Simple Mechanical Wave A single up-down motion applied to a taut string generates a pulse. The pulse then travels along the string at velocity v. Assumptions in this chapter: No friction-like forces within the string to dissipate wave motion. Strings are very long - no need to consider reflected waves from the far end. He moves his hand once.

jw Fundamentals of Physics 4 Traveling Waves Transverse Wave: The displacement (and velocity) of every point along the medium carrying the wave is perpendicular to the direction of the wave. Longitudinal Wave: The displacement (and velocity) of the element of the medium carrying the wave is parallel to the direction of the wave. e.g. a vibrating stringe.g. a sound wave They oscillate their hand in SHM. Longitudinal, Transverse and Mixed Type Waves

jw Fundamentals of Physics 5 Transverse Wave Displacement versus position not versus time Each point along the string just moves up and down. transverse wave applet

jw Fundamentals of Physics 6 amplitude - maximum displacement wavelength - distance between repetitions of the shape of the wave. angular wave number period - one full oscillation frequency - oscillations per unit time Wave Length & Frequency

jw Fundamentals of Physics 7 The phase, kx – wt, changes linearly with x and t, which causes the sine function to oscillate between +1 and –1. Wave Length & Frequency time space

jw Fundamentals of Physics 8 Each point on the wave, e.g. Point A, retains its displacement, y; hence: Speed of a Traveling Wave Consider a wave traveling in the positive x direction; the entire wave pattern moves a distance  x in time  t: Note: both x and t are changing! Differentiating:

jw Fundamentals of Physics 9 Direction of the Wave y = f (x + v t) traveling towards -x All traveling waves are functions of (kx +  t) = k(x + vt). Consider an unchanging pulse traveling along positive x axis. traveling towards +x y = f (x ’ ) = f (x - v t)

jw Fundamentals of Physics 10 Descriptions of the phase of a Traveling Wave

jw Fundamentals of Physics 11 Wave Speed on a Stretched String A single symmetrical pulse moving along a string at speed v wave. In general, the speed of a wave is determined by the properties of the medium through which it travels.

jw Fundamentals of Physics 12 Wave Speed on a Stretched String Consider a single symmetrical pulse moving along a string at speed v: Speed of a wave along a stretched string depends only on the tension and the linear density of the string and not on the frequency of the wave.  = tension in the string  = the string’s linear density Moving along with the pulse on the string. String moving to the left. (Roughly a circular arc)

jw Fundamentals of Physics 13 Energy of a Traveling String Wave Energy Driving force imparts energy to a string, stretching it. The wave transports the energy along the string.

jw Fundamentals of Physics 14 Driving force imparts energy to a stretched string. The wave transports energy along the string. –Kinetic energy - transverse velocity of string mass element,  m =   x –Potential energy - the string element  x stretches as the wave passes. Energy & Power of a Traveling String Wave (See Section 14-3)

jw Fundamentals of Physics 15 The Principle of Superposition for Waves y total (x,t) = y 1 (x,t) + y 2 (x,t) Overlapping waves algebraically add to produce a resultant wave: y 1 (x,t) y 2 (x,t) Add the amplitudes: Overlapping waves do not alter the travel of each other!

jw Fundamentals of Physics 16 The Principle of Superposition for Waves Interference of waves traveling in opposite directions. Constructive Interference Destructive Interference

jw Fundamentals of Physics 17 Interference of Waves Traveling in the Same Direction It is easy to show that:   = “phase difference” 

jw Fundamentals of Physics 18 Interference of Waves The magnitude of the resultant wave depends on the relative phases of the combining waves - INTERFERENCE. Constructive Interference Destructive InterferencePartial Interference

jw Fundamentals of Physics 19 Standing Waves y ’ (x,t) = y 1 (x,t) + y 2 (x,t) y 1 (x,t) = y m sin (k x -  t) y 2 (x,t) = y m sin (k x +  t) Two sinusoidal waves of the same amplitude and wavelength travel in opposite directions along a string: For a standing wave, the amplitude, 2y m sin(kx), varies with position. positive x direction Their interference with each other produces a standing wave: negative x direction It is easy to show that:

jw Fundamentals of Physics 20 Standing Wave The amplitude of a standing wave equals zero for: x = ½ n n = 0, 1, 2,... NODES

jw Fundamentals of Physics 21 Standing Waves x = ½ n n = 0, 1, 2,... NODES x = ½(n + ½) n = 0, 1, 2,... ANTINODES

jw Fundamentals of Physics 22 Reflections at a Boundary Reflected pulse has opposite sign Reflected pulse has same sign Newton’s 3 rd Law “soft” reflection “hard” reflection Tie the end of the string to the wall End of the string is free to move Antinode at boundary Node at boundary

jw Fundamentals of Physics 23 Reflections at a Boundary From high speed to low speed (low density to high density) From high density to low density

jw Fundamentals of Physics 24 Standing Waves & Resonance Resonance for certain frequencies for a string with both ends fixed. This can only be true when: v is the speed of the traveling waves on the string. Consider a string with both ends fixed; it has nodes at both ends. Only for these frequencies will the waves reflected back and forth be in phase.

jw Fundamentals of Physics 25 Standing Waves & Resonance A standing wave is created from two traveling waves, having the same frequency and the same amplitude and traveling in opposite directions in the same medium. Using superposition, the net displacement of the medium is the sum of the two waves. When 180° out-of-phase with each other, they cancel (destructive interference). When in-phase with each other, they add together (constructive interference).

jw Fundamentals of Physics 26 Standing Waves & Resonance The Harmonic Series both ends fixed

jw Fundamentals of Physics 27 String Fixed at One End fixed end Prenault’s applets free end Resonance: Standing wave applet

jw Fundamentals of Physics 28 Fundamentals of Physics Waves - II 1.Introduction 2.Sound Waves 3.The Speed of Sound 4.Traveling Sound Waves 5.Interference 6.Intensity & Sound Level The Decibel Scale 7.Sources of Musical Sound 8.Beats 9.The Doppler Effect Detector Moving; Source Stationary Source moving; Detector Stationary Bat Navigation 10.Supersonic Speeds; Shock Waves

jw Fundamentals of Physics 29 Sound Waves A sound wave is a longitudinal wave of any frequency passing through a medium (solid, liquid or gas).

jw Fundamentals of Physics 30 Elastic property of the medium –Strings - tension (  in N) –Sound - Bulk Modulus (B in N/m 2 ) Inertial property of the medium –Strings - linear mass density (  in kg/m) –Sound - volume mass density (  in kg/m 3 ) The Speed of Sound The speed of waves depends on the medium, not on the motion of the source.

jw Fundamentals of Physics 31 The Speed of Sound For an ideal gas, B/  can be shown to be proportional to absolute temperature; hence, the speed of sound depends on the square root of the absolute temperature. T = absolute temperature  = 1.4 for O 2 and N 2 (~air) R = “universal gas constant” = J/mol-K M = molar mass of the gas = 29 x kg/mol (for air) Equation for the speed of sound:

jw Fundamentals of Physics 32 Traveling Sound Waves Pressure-Variation Function: (pressure change as wave passes x) Displacement Function: ( of the air element about x) SHM

jw Fundamentals of Physics 33 Traveling Sound Waves As a sound wave moves in time, the displacement of air molecules, the pressure, and the density all vary sinusoidally with the frequency of the vibrating source. Slinky Demo

jw Fundamentals of Physics 34 Traveling Sound Waves As a sound wave moves in time, the displacement of air molecules, the pressure, and the density all vary sinusoidally with the frequency of the vibrating source.

jw Fundamentals of Physics 35 Interference Consider two sources of waves S 1 and S 2, which are “in phase”: “Constructive Interference”  is the “phase P 1 “arrive in phase”

jw Fundamentals of Physics 36 Interference Two sources of sound waves S 1 and S 2 : “Destructive Interference” arrive “out of phase”

jw Fundamentals of Physics 37 Psychological dimensions of sounds Pitch Loudness 1500 Hz 150 Hz 150 Hz with twice the amplitude of 1500 Hz A healthy young ear can hear sounds between ,000 Hz. - age reduces our hearing acuity for high frequencies. 300-Hz sound 500-Hz sound

jw Fundamentals of Physics 38 Intensity of a Sound Wave: The power of the wave is time rate of energy transfer. The area of the surface intercepting the sound. Intensity & Sound Level All the sound energy from the source spreads out radially and must pass through the surface of a sphere: In terms of the parameters of the source and of the medium carrying the sound, the sound intensity can be shown to be as follows:

jw Fundamentals of Physics 39 Intensity & Sound Level The Decibel Scale:  - Sound Level Mammals hear over an enormous range: where I 0 is the approximate threshold of human hearing. Sound level (or loudness) is a sensation in the consciousness of a human being. The psychological sensation of loudness varies approximately logarithmically; to produce a sound that seems twice as loud requires about ten times the intensity. (decibel) Alexander Graham Bell

jw Fundamentals of Physics 40 Intensity & Sound Level Every 10dB is a factor 10 change in intensity; 20 dB is a factor 100 change in intensity Human Perception of Sound ~3dB is a factor 2 change in intensity See Table 17-2.

jw Fundamentals of Physics 41 Intensity & Sound Level

jw Fundamentals of Physics 42 Sources of Musical Sound Closed End –Molecules cannot move Displacement node Pressure Antinode Open End –Molecules free to move Displacement Antinode Pressure Node Both ends closed  2 nodes with at least one antinode in between. Both ends open  2 antinodes with at least one node in between. One end closed  1 node and one antinode. Standing Waves in a Pipe

jw Fundamentals of Physics 43 Sources of Musical Sound Fundamental Frequency “1 st Harmonic” “Fundamental mode” nodes or antinodes at the ends of the resonant structure

jw Fundamentals of Physics 44 Sources of Musical Sound Both Ends OpenOne End Open Harmonic Number

jw Fundamentals of Physics 45 Sources of Musical Sound length of an instrument  fundamental frequency

jw Fundamentals of Physics 46 Sources of Musical Sound Fundamental & Overtones Overtones

jw Fundamentals of Physics 47 Beats 2 waves with slightly different frequencies are traveling to the right. The "beat" wave oscillates with the average frequency, and its amplitude envelope varies according to the difference frequency. The superposition of the 2 waves travels in the same direction and with the same speed.

jw Fundamentals of Physics 48 Beats Consider two similar sound waves: Superimpose them: “Beat Frequency”:

jw Fundamentals of Physics 49 Interference: Standing Wave created from two traveling waves: 2 waves with slightly different frequencies are traveling in the same direction. The superposition is a traveling wave, oscillating with the average frequency with its amplitude envelope varying according to the difference frequency. As the two waves pass through each other, the net result alternates between zero and some maximum amplitude. However, this pattern simply oscillates; it does not travel to the right or the left; it stands still. 2 sinusoidal waves having the same frequency (wavelength) and the same amplitude are traveling in opposite directions in the same medium. [one dot at an antinode and one at a node] Beats created from two traveling waves: Beats demo

jw Fundamentals of Physics 50 The Doppler Effect Doppler Effect Applet: Doppler Effect

jw Fundamentals of Physics 51 The Doppler Effect S - Wave SourceD - Detector (ear) Case 0: both are stationary frequency detected:

jw Fundamentals of Physics 52 The Doppler Effect Case 1: Detector is moving towards the source frequency detected:

jw Fundamentals of Physics 53 The Doppler Effect vD tvD tv tv t Case 1: Detector is moving towards the source Number of wavefronts intercepted “Rate of Interceptions” A higher frequency is detected See section 14.8

jw Fundamentals of Physics 54 The Doppler Effect Case 1: Detector is moving towards the source Applet: Doppler Effect Case 2: Detector is moving away from the source The detected frequency is less than the source frequency.

jw Fundamentals of Physics 55 The Doppler Effect Case 3: Source is moving towards the detector The detected frequency is greater than the source frequency. Case 4: Source is moving away from the detector

jw Fundamentals of Physics 56 Supersonic Speeds Applet: Doppler Effect

jw Fundamentals of Physics 57 Supersonic Speeds & Shock Waves No waves in front of the source. Waves pile up behind the source to form a shock wave. The “Mach Cone” narrows as v S goes up.

jw Fundamentals of Physics 58 Supersonic Speeds v S = Mach 1.8 You won’t hear it coming!

jw Fundamentals of Physics 59 Supersonic