Physics 1408-002 Principles of Physics Sung-Won Lee [email protected] Lecture 25 – Chapter 15 & 16 – April 21, 2009 Announcement I Lecture note is on the web Handout (6 slides/page) http://highenergy.phys.ttu.edu/~slee/1408/ *** Class attendance is strongly encouraged and will be taken randomly. Also it will be used for extra credits. HW Assignment #10 (Ch. 14,15,16) is placed on MateringPHYSICS, and is due by 11:59pm on Wednesday, 4/29 Exam III Average = 57 %!Alsinan 95!Cline 89!Rahnama 89!Malloch 86!Rudd 84!Matute 82!Whittle 82!Jorgenson 80!Scheers 80!Gilbreath 79!Grothe 79!Morales 79!Peterson 79!Rigsby 79!Germain A 78!Stephens 77!Aldughather76!Bradford 76!Deshpande 76!Ewing 76!Germain T 76!Announcements II Final Exam (All Chapters) 4/30 Thursday @ Sci 7 7:30 am – 10:00 am 2.5 hours Exams - Overview •! The exams are closed book. You may bring one hand-written 3” x 5” index card with formulae, etc. Telephones, iPod and other gizmos are not allowed. Small calculators are allowed. •! The final exam is comprehensive (3~5 min problems) and is a common exam for all sections (~Total 20~25 problems: i.e. 1~2 /chapter) – YES, it’s a multiple choice exam!! •! Please bring a scantron sheet (orange) for the exam!!Announcement III SI session by Reginald Tuvilla Next week, final exam review will be on Monday 04/06 from 4:30 - 7:30. (Room: HH 28) •! Characteristics & Types of Wave Motion •! Energy Transported by Waves •! Mathematical Representation of a Traveling Wave •! The Principle of Superposition, Reflection and Transmission •! Interference & Standing Waves Chapter 15 Wave Motion Characteristics of continuous wave moving through space: •! Amplitude, A •! Wavelength, ! •! Frequency, f and period, T •! Wave velocity, 15-1 Characteristics of Wave Motion 0( , ) sin 2x tD x t AT! "#$ %& '= ( +) *+ ,- ./ 022 angular frequency (rad/s)fT!" != = =2 wave number (rad/m)k!"# =2/ or 2v f k vkk! "# " "!$ %$ %= = = =& '& '( )( )[ ]0( , ) sinD x t A kx t! "= # +0Note that: (0,0) sinD A!=If x is fixed, D(x1,t) = A sin (kx1 - !t + ") gives a sinusoidal history graph at one point in space, x1. It repeats every Ts. Wave Equations Waves vs. Particles If two pitching machines simultaneously throw baseballs,"they will collide and bounce."Two particles cannot occupy the same space point at the same time.!Waves vs. Particles On the other hand, if two loud speakers make sound waves at the same time, they will pass through each other w/o collision. !Two waves can occupy the same space point at the same time.!•! If two or more traveling waves are moving through a medium, the resultant wave function is the algebraic sum of the wave functions of the individual waves 15-6 The Principle of Superposition Superposition of Sinusoidal Waves •! Assume two waves are traveling in the same direction, with the same frequency, wavelength and amplitude •! The waves differ in phase •! y1 = A sin(kx - !t) •! y2 = A sin(kx - !t + ") •! y = y1 + y2 = 2A cos("/2) sin(kx - !t + "/2) amplitude! How?!see next page!A Useful Trigonometric Identity sin a+sin b = 2 cos (( a – b ) / 2) sin (( a + b ) / 2) y1 = A sin (k x – ! t) y2 = A sin (k x – ! t + #) y = y1 + y2 = 2 A cos (# / 2) sin(k x – ! t + # / 2) AR a = k x – ! t b = k x – ! t + # amplitude!Superposition Example •! Two pulses are traveling in opposite directions; (a). The pulses have the same speed but different shapes !! When the waves start to overlap (b), the resultant wave function is y1 + y2 !! When crest meets crest (c) the resultant wave has a larger amplitude than either of the original waves !! The two pulses separate. They continue moving in their original directions. The shapes of the pulses remain unchanged;(d) 15-7 Reflection and Transmission When the pulse reaches the support, the pulse moves back along the string in the opposite direction. This is the reflection of thepulse. The pulse is inverted With a free end, the string is free to move vertically The pulse is reflected The pulse is not inverted 15-7 Reflection and Transmission •! Assume a light string is attached to a heavier string (see Fig.) •! Pulse travels through the light string and reaches the boundary •! The part of the pulse is inverted •! The reflected pulse has a smaller amplitude A wave encountering a denser medium will be partly reflected and partly transmitted; if the wave speed is less in the denser medium, the wavelength will be shorter.The superposition principle says that when two waves pass through the same point, the displacement is the sum of the individual displacements. In the figure below, (a) exhibits destructive interference and (b) exhibits constructive interference. 15-8 Interference Sinusoidal Waves with Constructive Interference •! When " = 0, then cos("/2) = 1 •! The amplitude of the resultant wave = 2A –! the resulting wave has an amplitude that is twice the amplitude of the individual waves •! The waves interfere constructively y = 2A cos("/2) sin(kx - !t + "/2) Total constructive interference y1 y2 y1 + y2 Total Constructive Interference Sinusoidal Waves with Destructive Interference •! When " = $ , then cos("/2) = 0 •! The amplitude of the resultant wave = 0 –! the resulting wave has zero amplitude the two waves completely cancel •! The waves interfere destructively y = 2A cos("/2) sin(kx - !t + "/2) Total destructive interference y1 y2 y1 + y2 Total destructive interference Sinusoidal Waves, General Interference •! When " is other than 0 or an even multiple of $, the amplitude of the resultant is between 0 and 2A y = 2A cos("/2) sin(kx - !t + "/2) Partially constructive interferencetotally constructive totally destructive!partially constructive!Sinusoidal Waves, Summary of Interference •! Constructive interference occurs when " = 0 –! Amplitude of the resultant = 2A •! Destructive interference occurs when " = n$ where n is an even integer –! Amplitude = 0 •! General interference occurs when 0 < " < n$ –! Amplitude = 0 < Aresultant < 2A Standing Waves Standing wave is a wave that remains in a constant position. (because it doesn’t seem to move) Standing waves occur when both ends of a string are fixed. There are nodes, where the amplitude is always zero, and antinodes, where the amplitude varies from zero to the maximum value. Standing