1. Stationary Waves Stationary waves are produced by superposition of two progressive waves of equal amplitude and frequency, travelling with the same speed in opposite directions. http://www2.biglobe.ne.jp/~norimari/science/JavaEd/e-wave4.html

2. Production of Stationary WavesStationary Waves A stationary wave would be set up by causing the string to oscillate rapidly at a particular frequency. If the signal frequency is increased further, overtone patterns appear.

3. Properties of a stationary wave (1) Stationary waves have nodes where there is no displacement at any time. In between the nodes are positions called antinodes, where the displacement has maximum amplitude. A vibrating loop N A N A N Vibrator

4. Properties of a stationary wave (2) The waveform in a stationary wave does not move through medium; energy is not carried away from the source. The amplitude of a stationary wave varies from zero at a node to maximum at an antinode, and depends on position along the wave.

5. Vibrations of particles in a stationary wave At t = 0, all particles are at rest because the particles reach their maximum displacements. At t = ¼ T, Particles a, e, and i are at rest because they are the nodes. Particles b, c and d are moving downward. They vibrate in phase but with different amplitude. Particles f, g and h are moving upward. They vibrate in phase but with different amplitude. t = 0 t = ¼T t = ⅜T t = ½T a b c d e f g h i a b c d e f g h i

6. Properties of a stationary wave (2) All particles between two adjacent nodes (within one vibrating loop) are in phase. Video 1.Stationary waves (string)Stationary waves (string) 2.Stationary waves (sound)Stationary waves (sound)

7. Modes of vibration of strings Picture of Standing Wave NameStructure 1st Harmonic or Fundamental 1 Antinode 2 Nodes 2nd Harmonic or 1st Overtone 2 Antinodes 3 Nodes 3rd Harmonic or 2nd Overtone 3 Antinodes 4 Nodes 4th Harmonic or 3rd Overtone 4 Antinodes 5 Nodes 5th Harmonic or 4th Overtone 5 Antinodes 6 Nodes L = ½λ 1 f 1 = v/2L L = λ 2 f 2 = v/L L = 1½λ 3 f 3 = 3v/2L L = 2λ 4 f 4 = 2v/L L = 2½λ 5 f 5 = 5v/2L http://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html L

8. Investigating stationary waves using sound waves and microwaves Moving the detector along the line between the wave source and the reflector enables alternating points of high and low signal intensity to be found. These are the antinodes and nodes of the stationary waves. The distance between successive nodes or antinodes can be measured, and corresponds to half the wavelength λ. If the frequency f of the source is known, the speed of the two progressive waves which produce the stationary wave can be obtained. Reflector Detector Wave source

9. Resonant Frequencies of a Vibrating String From the experiment, we find that There is a number of resonant frequencies in a vibrating string, The lowest resonant frequency is called the fundamental frequency (1 st harmonic), The other frequencies are called overtones (2 nd harmonic, 3 rd harmonic etc.), Each of the overtones has a frequency which is a whole-number multiple of the frequency of the fundamental.

10. Factors that determine the fundamental frequency of a vibrating string The frequency of vibration depends on the mass per unit length of the string, the tension in the string and, the length of the string. The fundamental frequency is given by where T = tension  = mass per unit length L = length of string

11. Vibrations in Air Column When a loudspeaker producing sound is placed near the end of a hollow tube, the tube resonates with sound at certain frequencies. Stationary waves are set up inside the tube because of the superposition of the incident wave and the reflected wave travelling in opposite directions. http://www.walter-fendt.de/ph11e/stlwaves.htm

12. Factors that determine the fundamental frequency of a vibrating air column The natural frequency of a wind instrument is dependent upon The type of the air column, The length of the air column of the instrument. Open tubeClosed tube

13. Name Modes of vibration for an open tube Picture of Standing WaveStructure 1st Harmonic or Fundamental 2 Antinodes 1 Node 2nd Harmonic or 1st Overtone 3 Antinodes 2 Nodes 3rd Harmonic or 2nd Overtone 4 Antinodes 3 Nodes 4th Harmonic or 3rd Overtone 5 Antinodes 4 Nodes 5th Harmonic or 4th Overtone 6 Antinodes 5 Nodes L = ½λ 1 f 1 = v/2L L = λ 2 f 2 = v/L L = 1½λ 3 f 3 = 3v/2L L = 2λ 4 f 4 =2v/L L = 2½λ 5 f 5 = 5v/2L

14. Modes of vibration for a closed tube Picture of Standing Wave NameStructure 1st Harmonic or Fundamental 1 Antinode 1 Node 3rd Harmonic or 1st Overtone 2 Antinodes 2 Nodes 5th Harmonic or 2nd Overtone 3 Antinodes 3 Nodes 7th Harmonic or 3rd Overtone 4 Antinodes 4 Nodes 9th Harmonic or 4th Overtone 5 Antinodes 5 Nodes L = ¼λ 1 f 1 = v/4L L = ¾λ 3 f 3 =3v/4L L = 1¼λ 5 f 5 =5v/4L L = 1¾λ 7 f 7 = 7v/4L L = 2¼λ 9 f 9 =9v/4L

15. The quality of sound (Timbre) The quality of sound is determined by the following factors: The particular harmonics present in addition to the fundamental vibration, The relative amplitude of each harmonic, The transient sounds produced when the vibration is started. 1 st overtone Fundamental 2 nd overtone 3 rd overtone resultant http://surendranath.tripod.com/Harmonics/Harmonics.html

16. Chladni ’ s Plate Chladni ’ s plate is an example of resonance in a plate. There are a number of frequencies at which the plate resonate. Each gives a different pattern.