Superposition & Interference

Lecture 29, Dec. 10 To do : Chapter 21 Understand beats as the superposition of two waves of unequal frequency. Prep for exam. Room 2103 Chamberlain Hall Sections: 602, 604, 605, 606, 610, 611, 612, 614 Room 5208 Sewell Social Science (1180 Observatory Dr.) Sections: 601, 603, 607, 608, 609, 613 Room 5310: McBurney and by prior arrangement 9:25 Evaluations Assignment HW12, Due Friday, Dec. 12 th, 10:59 PM Physics 207: Lecture 29, Pg 1 Superposition & Interference Consider two harmonic waves A and B meet at t=0. They have same amplitudes and phase, but ω 2 = 1.15 x ω 1. Beat Superposition The displacement versus time for each is shown below: A(ω 1 t) B(ω 2 t) C(t) = A(t) + B(t) CONSTRUCTIVE INTERFERENCE DESTRUCTIVE INTERFERENCE Physics 207: Lecture 29, Pg 2 Page 1

Consider A + B, Superposition & Interference D A (x,t)=a cos(k 1 x ω 1 t) D B (x,t)=a cos(k 2 x ω 2 t) And let x=0, D=D A +D B =2A cos[2π (f 1 f 2 )t/2] cos[2π (f 1 + f 2 )t/2] and f 1 f 2 f beat = = 1 / T beat A(ω 1 t) B(ω 2 t) t C(t) = A(t) + B(t) T beat Physics 207: Lecture 29, Pg 3 Exercise Superposition The traces below show beats that occur when two different pairs of waves are added (the time axes are the same). For which of the two is the difference in frequency of the original waves greater? A. Pair 1 B. Pair 2 C. The frequency difference was the same for both pairs of waves. D. Need more information. Physics 207: Lecture 29, Pg 4 Page 2

Organ Pipe Example A 0.9 m organ pipe (open at both ends) is measured to have it s first harmonic (i.e., its fundamental) at a frequency of 382 Hz. What is the speed of sound (refers to energy transfer) in this pipe? L=0.9 m f = 382 Hz and f λ = v with λ = 2 L / m (m = 1) v = 382 x 2(0.9) m v = 687 m/s Physics 207: Lecture 29, Pg 5 Standing Wave Question What happens to the fundamental frequency of a pipe, if the air (v =300 m/s) is replaced by helium (v = 900 m/s)? Recall: f λ = v (A) Increases (B) Same (C) Decreases Physics 207: Lecture 29, Pg 6 Page 3

Sample Problem The figure shows a snapshot graph D(x, t = 2 s) taken at t = 2 s of a pulse traveling to the left along a string at a speed of 2.0 m/s. Draw the history graph D(x = 2 m, t) of the wave at the position x = 2 m. Physics 207: Lecture 29, Pg 7 Sample Problem History Graph: 2-2 2 3 4 5 6 7 time (sec) Physics 207: Lecture 29, Pg 8 Page 4

Example problem Two loudspeakers are placed 1.8 m apart. They play tones of equal frequency. If you stand 3.0 m in front of the speakers, and exactly between them, you hear a maximum of intensity. As you walk parallel to the plane of the speakers, staying 3.0 m away, the sound intensity decreases until reaching a minimum when you are directly in front of one of the speakers. The speed of sound in the room is 340 m/s. a. What is the frequency of the sound? b. Draw, as accurately as you can, a wave-front diagram. On your diagram, label the positions of the two speakers, the point at which the intensity is maximum, and the point at which the intensity is minimum. c. Use your wave-front diagram to explain why the intensity is a minimum at a point 3.0 m directly in front of one of the speakers. Physics 207: Lecture 29, Pg 9 Example problem Two loudspeakers are placed 1.8 m apart. They play tones of equal frequency. If you stand 3.0 m in front of the speakers, and exactly between them, you hear a maximum of intensity. As you walk parallel to the plane of the speakers, staying 3.0 m away, the sound intensity decreases until reaching a minimum when you are directly in front of one of the speakers. The speed of sound in the room is 340 m/s. What is the frequency of the sound? v = f λ but we don t know f or λ DRAW A PICTURE Constructive Interference in-phase Destructive Interference out-of-phase Physics 207: Lecture 29, Pg 10 Page 5

Example problem Two loudspeakers are placed 1.8 m apart. They play tones of equal frequency. If you stand 3.0 m in front of the speakers, and exactly between them, you hear a maximum of intensity. v = 340 m/s. PUT IN GEOMETRY v = f λ but we don t know f or λ AC - BC = 0 (0 phase differenc) AD - BD = λ/2 (π phase shift) AD = (3.0 2 +1.8 2 ) 1/2 A BD = 3.0 λ = 2(AD-BD) =1.0 m Destructive B Interference out-of-phase 1.8 m 3.0 m Constructive Interference in-phase D C Physics 207: Lecture 29, Pg 11 Example problem Two loudspeakers are placed 1.8 m apart. They play tones of equal frequency. If you stand 3.0 m in front of the speakers, and exactly between them, you hear a maximum of intensity. b. Draw, as accurately as you can, a wave-front diagram. On your diagram, label the positions of the two speakers, the point at which the intensity is maximum, and the point at which the intensity is minimum. c. Use your wave-front diagram to explain why the intensity is a minimum at a point 3.0 m directly in front of one of the speakers. Physics 207: Lecture 29, Pg 12 Page 6

Sample problem A tube, open at both ends, is filled with an unknown gas. The tube is 190 cm in length and 3.0 cm in diameter. By using different tuning forks, it is found that resonances can be excited at frequencies of 315 Hz, 420 Hz, and 525 Hz, and at no frequencies in between these. a. What is the speed of sound in this gas? b. Can you determine the amplitude of the wave? If so, what is it? If not, why not? Physics 207: Lecture 29, Pg 13 Sample problem A tube, open at both ends, is filled with an unknown gas. The tube is 190 cm in length and 3.0 cm in diameter. By using different tuning forks, it is found that resonances can be excited at frequencies of 315 Hz, 420 Hz, and 525 Hz, and at no frequencies in between these. What is the speed of sound in this gas? L=1.9 meters and f m = vm/2l 315 = v m/2l 420 = v (m+1)/2l 525 = v (m+2)/2l v/2l = 105 v = 3.8 x 105 m/s =400 m/s b) Can you determine the amplitude of the wave? If so, what is it? If not, why not? Answer: No, the sound intensity is required and this is not known. Physics 207: Lecture 29, Pg 14 Page 7

Sample Problem The picture below shows two pulses approaching each other on a stretched string at time t = 0 s. Both pulses have a speed of 1.0 m/s. Using the empty graph axes below the picture, draw a picture of the string at t = 4 s. Physics 207: Lecture 29, Pg 15 An example A heat engine uses 0.030 moles of helium as its working substance. The gas follows the thermodynamic cycle shown. a. Fill in the missing table entries b. What is the thermal efficiency of this engine? c. What is the maximum possible thermal efficiency of an engine that operates between T max and T min? A: T= 400 K B: T=2000 K C: T= 1050 K B A B B C C A NET Q H 600 J 0 J 0 J 600 J Q L 0 J 0 J W by 0 J -161 J E Th 600 J -243 J A C Physics 207: Lecture 29, Pg 16 Page 8

The Full Cyclic Process A heat engine uses 0.030 moles of helium as its working substance. The gas follows the thermodynamic cycle shown. a. What is the thermal efficiency of this engine? b. What is the maximum possible thermal efficiency of an engine that operates between T max and T min? (T = pv/nr, pv γ =const.) A: T= 400 K=1.01x10 5 10-3 / 0.030/8.3 B: T=2000 K p B V Bγ = p C V C γ B (p B V B γ / p C ) 1/γ = V C (5 (10-3 ) 5/3 / 1) 3/5 = V C = 2.6 x 10-3 m 3 C: T= 1050 K A C Physics 207: Lecture 29, Pg 17 The Full Cyclic Process A: T= 400 K=1.01x10 5 10-3 / 0.030/8.3 B: T=2000 K Q = n C v T = 0.030 x 1.5 x 8.3 T V C = 2.6 x 10-3 m 3 Q = 0.3735 T C: T= 1050 K W C A (by) = p V = 1.01x10 5 1.6 x 10-3 J A B B C C A NET Q H 600 J 0 J 0 J 600 J Q L 0 J 0 J 404 J 404 J W by 0 J 355 J -161 J 194 J E Th 600 J -355 J -243 J 0 J B η = W by / Q H = 194/ 600 = 0.32 η Carnot = 1- T L / T H = 1-400/2000 =0.80 A C Physics 207: Lecture 29, Pg 18 Page 9

An example A monatomic gas is compressed isothermally to 1/8 of its original volume. Do each of the following quantities change? If so, does the quantity increase or decrease, and by what factor? If not, why not? a. The rms speed v rms b. The temperature c. The mean free path d. The molar heat capacity C V Physics 207: Lecture 29, Pg 19 An example A creative chemist creates a small molecule which resembles a freely moving bead on a wire (rotaxanes are an example). The wire is fixed and the bead does not rotate. If the mass of the bead is 10-26 kg, what is the rms speed of the bead at 300 K? Below is a rotaxane model with three beads on a short wire Physics 207: Lecture 29, Pg 20 Page 10

An example A creative chemist creates a small molecule which resembles a freely moving bead on a wire (rotaxanes are an example). Here the wire is a loop and rigidly fixed and the bead does not rotate. If the mass of the bead is 10-26 kg, what is the rms speed of the bead at 300 K? Classically there is ½ k B T of thermal energy per degree of freedom. Here there is only one so: ½ mv rms2 = ½ k Boltzmann T v rms =(k Boltzmann T/m) ½ = 640 m/s NOT for any exam but an interesting aside: If a molecular wire has a length of 20 nm and will break if the force exceeds 10-10 N, estimate the temperature at which it will break. (I get, 36000 K.covalent chemical bonds are very strong) Physics 207: Lecture 29, Pg 21 An example A small speaker is placed in front of a block of mass 4 kg. The mass is attached to a Hooke s Law sping with sping constant 100 N/m. The mass and speaker have a mechanical energy of 200 J and are undergoing one dimensional simple harmonic motion. The speaker emits a 200 Hz tone. For a person is standing directly in front of the speaker, what range of frequencies does he/she hear? The closest the speaker gets to the person is 1.0 m. By how much does the sound intensity vary in terms of the ratio of the loudest to the softest sounds? Physics 207: Lecture 29, Pg 22 Page 11

An example A small speaker is placed in front of a block of mass 4 kg. The mass is attached to a Hooke s Law sping with sping constant 100 N/m. The mass and speaker have a mechanical energy of 200 J and are undergoing one dimensional simple harmonic motion. The speaker emits a 200 Hz tone. (v sound = 340 m/s) For a person is standing directly in front of the speaker, what range of frequencies does he/she hear? E mech =200 J = ½ mv max2 = ½ ka 2 v max =10 m/s Now use expression for Doppler shift where v source = + 10 m/s and 10 m/s Physics 207: Lecture 29, Pg 23 An example A small speaker is placed in front of a block of mass 4 kg. The mass is attached to a Hooke s Law sping with sping constant 100 N/m. The mass and speaker have a mechanical energy of 200 J and are undergoing one dimensional simple harmonic motion. The speaker emits a 200 Hz tone. (v sound = 340 m/s) The closest the speaker gets to the person is 1.0 m. By how much does the sound intensity vary in terms of the ratio of the loudest to the softest sounds? E mech =200 J = ½ mv max2 = ½ ka 2 A = 2.0 m Distance varies from 1.0 m to 1.0+2A or 5 meters where P is the power emitted Ratio I loud /I soft = (P/4π r loud2 )/(P/4π r soft2 ) = 5 2 /1 2 = 25 Physics 207: Lecture 29, Pg 24 Page 12

An example problem In musical instruments the sound is based on the number and relative strengths of the harmonics including the fundamental frequency of the note. Figure 1a depicts the first three harmonics of a note. The sum of the first two harmonics is shown in Fig. 1b, and the sum of the first 3 harmonics is shown in Fig. 1c. Which of the waves shown has the shortest period? a. 1 st Harmonic b. 2 nd Harmonic c. 3 rd Harmonic d. Figure 1c At the 2 nd position(1 st is at t=0) where the three curves intersect in Fig. 1a, the curves are all: a. in phase b. out of phase c. at zero displacement d. at maximum displacement a displacement b displacement c displacement time time time Figure Physics 1: 207: Wave Lecture superposition 29, Pg 25 An example problem The frequency of the waveform shown in Fig. 1c is a. the same as that of the fundamental b. the same as that of the 2 nd harmonic c. the same as that of the 3 rd harmonic d. sum of the periods of the 1 st, 2 nd & 3 rd harmonics Which of the following graphs most accurately reflects the relative amplitudes of the harmonics shown in Fig. 1? a b c amplitude amplitude 1 2 3 harmonic 1 2 3 harmonic amplitude d amplitude 1 2 3 harmonic 1 2 3 harmonic a displacement b displacement c displacement time time time Figure Physics 1: 207: Wave Lecture superposition 29, Pg 26 Page 13

Chapter 1 Physics 207: Lecture 29, Pg 27 Important Concepts Physics 207: Lecture 29, Pg 28 Page 14

Chapter 2 Physics 207: Lecture 29, Pg 29 Chapter 3 Physics 207: Lecture 29, Pg 30 Page 15

Chapter 4 Physics 207: Lecture 29, Pg 31 Chapter 4 Physics 207: Lecture 29, Pg 32 Page 16

Chapter 5 Physics 207: Lecture 29, Pg 33 Chapter 5 & 6 Physics 207: Lecture 29, Pg 34 Page 17

Chapter 6 Chapter 7 Physics 207: Lecture 29, Pg 35 Chapter 7 Physics 207: Lecture 29, Pg 36 Page 18

Chapter 7 (Newton s 3 rd Law) & Chapter 8 Physics 207: Lecture 29, Pg 37 Chapter 9 Chapter 8 Physics 207: Lecture 29, Pg 38 Page 19

Chapter 9 Physics 207: Lecture 29, Pg 39 Chapter 10 Physics 207: Lecture 29, Pg 40 Page 20

Chapter 10 Physics 207: Lecture 29, Pg 41 Chapter 10 Physics 207: Lecture 29, Pg 42 Page 21

Chapter 11 Physics 207: Lecture 29, Pg 43 Chapter 11 Physics 207: Lecture 29, Pg 44 Page 22

Chapter 12 and Center of Mass Physics 207: Lecture 29, Pg 45 Chapter 12 Physics 207: Lecture 29, Pg 46 Page 23

Important Concepts Physics 207: Lecture 29, Pg 47 Chapter 12 Physics 207: Lecture 29, Pg 48 Page 24

Angular Momentum Physics 207: Lecture 29, Pg 49 Hooke s Law Springs and a Restoring Force Key fact: ω = (k / m) ½ is general result where k reflects a constant of the linear restoring force and m is the inertial response (e.g., the physical pendulum where ω = (κ / I) ½ Physics 207: Lecture 29, Pg 50 Page 25

Simple Harmonic Motion Maximum potential energy Maximum kinetic energy Physics 207: Lecture 29, Pg 51 Resonance and damping Energy transfer is optimal when the driving force varies at the resonant frequency. Types of motion Undamped Underdamped Critically damped Overdamped Physics 207: Lecture 29, Pg 52 Page 26

Fluid Flow Physics 207: Lecture 29, Pg 53 Density and pressure Physics 207: Lecture 29, Pg 54 Page 27

Response to forces States of Matter and Phase Diagrams Physics 207: Lecture 29, Pg 55 Ideal gas equation of state Physics 207: Lecture 29, Pg 56 Page 28

pv diagrams Thermodynamics Physics 207: Lecture 29, Pg 57 Work, Pressure, Volume, Heat T can change! Physics 207: Lecture 29, Pg 58 Page 29

Chapter 18 Physics 207: Lecture 29, Pg 59 Thermal Energy Physics 207: Lecture 29, Pg 60 Page 30

Relationships Physics 207: Lecture 29, Pg 61 Chapter 19 Physics 207: Lecture 29, Pg 62 Page 31

Refrigerators Physics 207: Lecture 29, Pg 63 Carnot Cycles Physics 207: Lecture 29, Pg 64 Page 32

Work (by the system) Physics 207: Lecture 29, Pg 65 Chapter 20 Physics 207: Lecture 29, Pg 66 Page 33

Displacement versus time and position Physics 207: Lecture 29, Pg 67 Sinusoidal Waves (Sound and Electromagnetic) Physics 207: Lecture 29, Pg 68 Page 34

Doppler effect Physics 207: Lecture 29, Pg 69 Chapter 21 Physics 207: Lecture 29, Pg 70 Page 35

Standing Waves Physics 207: Lecture 29, Pg 71 f D(0, t) = 2 A cos( 2π f 1 f 2 f beat = 1/ T Beats 1 f 2 beat 2 t) cos( 2π f + f 1 2 2 f + f 1 2 f 2 t) avg = 1/ T avg Physics 207: Lecture 29, Pg 72 Page 36

Lecture 29, Dec. 10 Assignment HW12, Due Friday, Dec. 12 th I hope everyone does well on their finals! Have a great break! Physics 207: Lecture 29, Pg 73 Page 37