Physics 231 Lecture 25

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1 Physics 231 Lecture 25 Spherical waves P Main points of today s I = lecture: 2 4πr Wave Dopper speed shift for a string v + v o ƒ' = ƒ F v = v vs µ Interference of sound waves L Here F is the string tension 1 L2 = nλ constructive (n = 0,1,2) L and µ is the mass/length. 1 L2 = ( n + 1/ 2) λ destructive Superposition Standing waves and on interference string: of waves; wave forms interfere. nv 2L f n = Opp. λn = sign destructive. n = 1,2,3,,, 2L n Same sign constructive Standing wave in air columns: Reflection of waves. both ends open nvrigid end 2L= inverted wave f n = λn = n = 1, 2, 3,,, 2Lfree end n= non-inverted wave one end open nv 4L f n = λn = n = 13,, 5,,, 4L n Beat Phenomena: f beat = f 1 f 2

2 Human Hearing Human ears can detect sound waves in the frequency range from 20 to 20,000 Hz Corresponding range of wavelengths: β [db] 80 Teenager I min [W] m to m β = 10log 10 I I Retiree f [Hz] You are most sentive to sounds in mid higher frequencies. Your ears become less sensitive as you age. I 0 I min

The sound wave power P crossing point B and point C per unit time are the same.

3 Propagation of spherical and plane waves. If one vibrates as piston in a tube of cross-sectional area A, the sound waves travel down the tube in a straight line, like a wave on a string. The sound wave power P crossing point B and point C per unit time are the same. Since the area A of the tube is constant, the intensity I=P/A is the same at both points. If one suspends non-directional sound source with power P in air, the sound will radiate in all directions. All points on a sphere of radius r will see the same sound intensity: I = P A = P P = I A = const. 4πr 2 B C r Fig 14.4, p. Slide 8

4 Example When a helicopter is hovering 1100 m directly overhead, an observer on the ground measures a sound intensity I 1. Assume that sound is radiated uniformly as a spherical wave from the helicopter and that ground reflections are negligible. How far must the helicopter fly in a straight line parallel to the ground before the observer measures a sound intensity of I 2 =I 1 /5? z y x r 2 d h I I I 2 1 = 1/5 = P I = P 1 4πh πr2 r = h + d = 5h 2 2 d = 4h d = 2h = 2200m

Doppler effect for moving observer An observer is moving toward a stationary source Due to his movement, the observer detects an additional number of wave fronts

5 Doppler effect for moving observer An observer is moving toward a stationary source Due to his movement, the observer detects an additional number of wave fronts The frequency heard is increased An observer is moving away from a stationary source The observer detects fewer wave fronts per second The frequency appears lower

6 Doppler Effect, Source in Motion general formula ƒ' = ƒ v v + v v o s Both the source and the observer could be moving As the source moves toward the observer (A), the wavelength appears shorter and the frequency increases As the source moves away from the observer (B), the wavelength appears longer and the frequency appears to be lower Use positive values of v o and v s when the motion is toward Frequency appears higher Use negative values of v o and v s when the motion is away Frequency appears lower

7 Example A train at rest emits a sound at a frequency of Hz. An observer in a car travels away from the sound source at a speed of 30.0 m/s. What is the frequency heard by the observer? (assume the sound velocity v=343 m/s) a) 513 Hz b) 713 Hz c) 913 Hz d) 1013 Hz e) 1113 Hz vs = 0, ( + ) v v ' f = o v f vo = 30m / s 313 = 1000Hz = 913Hz 343 ƒ' = ƒ v v + v v o s

8 Conceptual quiz Three observers,a,b, and C are listening to a moving source of sound. The diagram below shows the location of the wavecrests of the moving source with respect to the three observers.which of the following is true? a. The wavefronts move faster at A than at B and C. Wavefronts move b. The wavefronts move faster at C than at A and B. at sound speed 343 m/s c. The frequency of the sound is highest at A. d. The frequency of the sound is highest at B. e. The frequency of the sound is highest at C. f = v sound λ λ is smallest at C f is largest at C

9 Example Two trucks travel at the same speed. They are far apart on adjacent lanes and approach each other essentially head-on. One driver hears the horn of the other truck at a frequency that is 1.2 times the frequency he would hear if the trucks were stationary. The speed of sound is 343 m/s. At what speed is each truck moving? ' f givens: 1.2, v v v v f = = = = f ' = ( v+ vo ) ( v+ v ) f = ( v vs ) ( v v ) ( + ) 1.2 ( ) f ' v v = = f v v v + v = 1.2 v v ( ) s o truck 1.2v v = v + 1.2v 0.2v = 2.2v v = v = i 343m / s ans. v = 31m / s f = 1.2v 1.2v

10 Relation between amplitude and intensity A -A x!me (s) For sound, the intensity I is propor!onal to the square of the amplitude of the longitudinal wave I~A 2 Amplitudes are s!ll important.

11 Interference of sound wave amplitudes Assume both speakers are connected to the same signal generator. Then these sound waves interfere as follows: Constructive interference occurs when the path difference between two waves motion is zero or some integer multiple of wavelengths path difference = r 1 -r 2 = nλ Destructive interference occurs when the path difference between two waves motion is an odd half wavelength path difference = r 1 -r 2 = (n + ½)λ

12 Sound Interference in Space 2 speakers producing same tone Point A: maximum from both sound waves Constructive Interference A B C Point B: maximum from 1, minimum from 2 Destructive Interference

13 Path length difference, Δr Sound Interference in Space Condition for constructive interference: Path length difference between the two paths is an integer multiple of the wavelength. Δr = nλ; for all n = 0,1,2,3,... constructive interference Condition for destructive interference: Path length difference between the two paths is a odd half-integer multiple of the wavelength. Δr = (n )λ; for all n = 0,1,2,3,... destructive interference

14 Points of constructive interference lie on (almost straight) lines Between them are lines of destructive interference Sound Interference in Space

Interference example If both speakers are driven with the signal generator, what kind of interference will be along the center line passing through O a) Constructive b) Destructive If point P is the

15 Interference example If both speakers are driven with the signal generator, what kind of interference will be along the center line passing through O a) Constructive b) Destructive If point P is the first interference minimum as one moves away from O, r 1 -r 2 = a) 0 b) λ/2 c) 3λ/2 d) λ e) 2 λ If the r 1 -r 2 = λ/2, the pressure maxima from one speaker arrive at the same time as the pressure minima from the other. Thus, there no so strong pressure changes with time at that point; the sound intensity is a minimum. That also occurs when r 1 -r 2 = 3λ/2, but that is not the first minimum.

16 Beats Beats are alternations in loudness, due to interference Waves have slightly different frequencies and the time between constructive and destructive interference alternates f beat = f 1 f 2

17 Conceptual quiz The traces below show beats that occur when two different pairs of waves are added. For which of the two is the difference in frequency of the original waves greater? 1. Pair 1 2. Pair 2 3. The frequency difference was the same for both pairs of waves. 4. Need more information. t t f beat = f 1 f 2

18 Example Each of three tuning forks A, B, and C has a slightly different frequency. When A and B are sounded together, they produce a beat frequency of 2 Hz. When A and C are sounded together, the produce a beat frequency of 5 Hz. What is the beat frequency that occurs when B and C are sounded together? There are two possible answers. if fa fb fc = = = 100Hz, 98 or 102 Hz 95 or 105 Hz fbeat,b C = fb fc = 3 or 7

Physic 231 Lecture 35

Physic 231 Lecture 35 Physic 31 Lecture 35 Main points o last lecture: Waves transverse longitudinal traveling waves v wave λ Wave speed or a string v F µ Superposition and intererence o waves; wave orms interere. Relection