LECTURE 5 STANDING WAVES & BEATS. Instructor: Kazumi Tolich

Transcription

1 LECTURE 5 STANDING WAVES & BEATS Instructor: Kazumi Tolich

2 Lecture 5 2 Reading chapter 16-1 to 16-2 Superposition of waves Interference of harmonic waves Standing waves

3 Principle of Superposition 3 When two or more waves are simultaneously at a single point in space, the displacement of the medium at that point is the sum of the displacements due to each individual wave. If the amplitude grows too large, superposition may fail. Also, real waves do not have kinks.

4 Quiz:1

5 Beats 5 The interference of two sound waves with slightly different frequencies, f 1 and f 2, produces beats. The amplitude of the resulting wave is modulated. Long lines The beat frequency is given by Short lines

6 Quiz: 2

7 Example: 1 7 One flutist plays a note of f 1 = 510 Hz, while a second flutist plays a note of f 2 = 512 Hz. What frequency do you hear? What is the beat frequency?

8 Demo 1 8 Speaker and Oscillator(s) With one oscillator: demonstration of frequency and pitch; human hearing range ~20Hz to ~20000Hz. With two oscillators: demonstration of beats due to two oscillators with different frequencies. Beats (Tuning Forks) Demonstration of beats due to two tuning forks with different frequencies.

9 Phase difference due to path difference 9 A phase difference between two waves is often the result of a difference in path length, Δr. 4λ Constructive interference occurs when δ = Nπ, where N = 0, 2, 4, Destructive interference occurs when δ = Nπ, where N = 1, 3, 5, 5λ 4λ 4.5λ

10 Demo 2 10 Two Speaker Interference Demonstration of constructive and destructive interference due to path differences.

11 Example 2 11 Two audio speakers facing in the same direction oscillate in phase at the same frequency. They are separated by a distance equal to one-third of a wavelength. Point P is in front of both speakers, on the line that passes through their centers. The amplitude of the sound at P due to either speaker acting alone is A. What is the amplitude (in terms of A) of the resultant wave at point P?

12 Standing waves & resonant frequencies 12 If waves are confined in space, reflections at both ends cause the waves to travel in in opposite directions and interfere. For a given string or pipe, there are certain frequencies for which superposition results in a stationary vibration pattern called a standing wave. Frequencies that produce standing waves are called resonant frequencies of the string system. The lowest resonant frequency is called fundamental, or first harmonic. Then the higher ones are 2nd harmonic, 3rd harmonic, etc.

13 13 Nodes and antinodes Nodes are spaced a distance λ/2 apart, and they include the point at which the string is anchored to the wall. Nodes are points of maximum destructive interference. Antinodes are spaced λ/2 apart also. They are points of maximum constructive interference. y or s

14 Standing waves on a string (fixed-fixed) 14 The standing wave condition and resonant frequency for the nth harmonic for a string with length L, and both ends fixed are L

15 Standing wave on a string (fixed-driven) 15 The standing wave condition for nth harmonic for a string with length L, and one end fixed and the other end connected to a vibrator is

16 Demo 3 16 Standing Waves in Rubber Tubing (Vary Frequency) When the right frequencies are reached, the tubing vibrates in various standing wave modes.

17 Standing wave on a string (fixed-free) 17 The standing wave condition and resonant frequency for the nth harmonic for a string with length L, and one end fixed and the other end free are The free end must be at antinode so that the string meets the boundary condition.

18 Sound waves in a pipe (closed-closed) 18 A closed end causes a node there due to the boundary conditions. Physical representation of the n = 2 mode.

19 Sound waves in a pipe (open-open) 19 node

20 20 Sound waves in a pipe (open-closed)

21 Pipes and modes summary 21 n = 1 n = 1 n = 1 n = 2 n = 2 n = 3 n = 3 n = 3 n = 5 Open-Open or Closed-Closed Open-Closed

22 Demo 4 & 5 22 Open and Closed End Pipes Various pipes with different resonant frequencies. Beats (Singing Pipes) Demonstration of resonance and beats due to difference in the resonant frequencies of two pipes.

23 Example 3 23 A shower stall is L = 2.45 m tall. Assuming the shower stall is a closed-closed pipe, for what frequencies less than 500 Hz can there be vertical standing sound waves in the shower stall? Assume the speed of sound is v = 343 m/s.

24 24 Quiz: 3

25 Pop music 25 When air is blown across the open top of a pop bottle, the turbulent air flow can cause an audible standing wave. The standing wave will have an antinode at the top and a node at the bottom.