Physics 231 Lecture 28

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Primrose Bryan
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1 Physics 231 Lecture 28 Main points of today s lecture: Reflection of waes. rigid end inerted wae free end non-inerted wae Standing waes on string: n 2L f n λn n 1, 2, 3,,, 2L n Standing wae in air columns: f n both ends open n 2L λ n 2L n one end open n 4L f n λn n 13,, 5,,, 4L n Resonance n 1,2,3,,,

2 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

3 reflection of wae at a boundary amplitude at fixed end is zero and reflected wae is inerted. Amplitude at free end is double and reflected wae is same polarity (in this case it is positie.

4 Superposition principle Two traeling pulse waes pass through each other without affec4ng each other. The resul4ng displacement is the superposi4on of the two indiidual waes.

Anti-nodes are the points where fundamental the motion is maximal. 1 st harmonic Motion with one antinode has λ/2 L, two antinodes has λl.

5 Standing waes on a string Two traeling sinusoidal waes of the same frequency moing in opposite directions can superimposed so as to make a standing wae: Because the string is held at each end, those points are nodes of the oscillator motion, where the amplitude of motion anishes. Anti-nodes are the points where fundamental the motion is maximal. 1 st harmonic Motion with one antinode has λ/2 L, two antinodes has λl... The general relationship is: 2L n n F λ n ; f n n 1,2,3,,, n λ 2L 2L µ n higher harmonics n1 fund. E n2 2 nd Har. E n3 3 d Har. B n4 4 th Har. E n5 5 th Har. G #

6 Example A stretched string fixed at each end has a mass of 40.0 g and a length of 8.00 m. The tension in the string is 49.0 N. (a) Determine the positions of the nodes and antinodes for the 2 nd harmonic. (b) What is the ibration frequency for this harmonic? Second harmonic (n 0,4, and 8 m and anti-nodes at 2 and 6 m. f 2 2 2L F m/l 2) has nodes at 49N 0.04kg / 8m 98m / s Fig 14.18, p. 443 Slide 25 ans. f 2 λ 2 L 98m / s 8m 12.4Hz

7 Standing waes in an air columns: both ends open If one end of the air column is closed, a node must exist at this end since the moement of the air is restricted If the end is open, the elements of the air hae complete freedom of moement and an antinode exists n 2L f n λn n 1, 2, 3,,, 2L n

8 Interference: Standing Wae in a open pipe If two waes trael in opposite direc4ons and 12, the superposi4on of the two waes produces a standing wae: maxima and minima always appear at the same loca4on

9 Standing waes in an air columns: one end closed If one end of the air column is closed, a node must exist at this end since the moement of the air is restricted If the end is open, the elements of the air hae complete freedom of moement and an antinode exists n 4L f n λn n 13,, 5,,, 4L n

10 Quiz The fundamental frequencies of two air columns are the same. Column A is open at both ends, while column B is open at only one end. The length of column A is 0.60 m. What is the length of column B? a) 0.1 m b) 0.2 m c) 0.3 m d) 0.5 m e) 0.6 m f A f B fa 2L A fb 4L B fb fa 4L B 2L 4L A B L L /2 B A ans. LB 0.3m 2L A

11 Mach Cone What happens when we exceed the speed of sound with our source elocity? source < sound source > sound

12 Mach Cone µ

13 Mach Cone What happens when we exceed the speed of sound with our source elocity? Mach Cone Mach angle: sound Δt θ M sinθ M soundδt source Δt source Δt θ M sin 1 sound source Physics 231C 74

Mach Cone A apor cone is formed as this F/A-18F Super Hornet closes on Mach 1.(Hayman Tam) http://inflight.

14 Mach Cone A apor cone is formed as this F/A-18F Super Hornet closes on Mach 1.(Hayman Tam)

15 Forced ibrations and resonance A system with a driing force will force a ibration at its frequency When the frequency of the driing force equals the natural frequency of the system, the system is said to be in resonance

16 Conceptual question A string is clamped at both ends and plucked so it ibrates in a standing mode between two extreme positions a and b. Let upward motion correspond to positie elocities.when the string is in position c, the instantaneous elocity of points along the string: a. is zero eerywhere. b. is positie eerywhere. c. is negatie eerywhere. d. depends on location.

Lecture 18. Sound Waves: Intensity, Interference, Beats and Doppler Effect.

Lecture 18. Sound Waves: Intensity, Interference, Beats and Doppler Effect. Lecture 18 Sound Waes: Intensity, Interference, Beats and Doppler Effect. Speed of sound Speed of soun in air, depends on temperature: = (331 + 0.60 T ) m/s where T in C Sound intensity leel β = 10log