LECTURE 7 STANDING WAVES

Instructor: Kazumi Tolich Lecture 7

2

¨ Reading chapter 16-2 ¤ Standing waves Phase difference due to path difference

3 4λ ¨ A phase difference between two waves is often the result of a difference in path length, Δr.

5λ ¨ Constructive interference occurs when δ = Nπ, where N = 0, 2, 4, …

¨ Destructive interference occurs when δ = Nπ, where N = 1, 3, 5, …

4λ

4.5λ Demo 1

4

¨ Two Speaker Interference ¤ Demonstration of constructive and destructive interference due to path length differences. Quiz: 1

5 Example 1

6

¨ 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? Standing waves & resonant frequencies

7

¨ 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. Nodes and antinodes

8

¨ 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 y or s maximum destructive interference. ¨ Antinodes are spaced λ/2 apart also. They are points of maximum constructive interference. Standing waves on a string (fixed-fixed)

9

¨ The standing wave condition and resonant frequency for the nth harmonic for a string with length L, and both ends fixed are

L Standing wave on a string (fixed-driven)

10

¨ 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 Demo 2

11

¨ Standing Waves in Rubber Tubing (Vary Frequency) ¤ When the right frequencies are reached, the tubing vibrates in various standing wave modes. Standing wave on a string (fixed-free)

12

¨ 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. Sound waves in a pipe (closed-closed)

13

¨ A closed end causes a node there for the displacement wave due to the boundary conditions.

Physical representation of the n = 2 mode. Sound waves in a pipe (open-open)

14

node Sound waves in a pipe (open-closed)

15 Pipes and modes summary

16

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 Example 2

17

¨ 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. Quiz: 2

18 Demo 3 & 4

19

¨ 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. Pop music

20

¨ 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.