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Standing Waves 1 Part 1: Strings (Transverse Standing Waves) 05/03/08

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Standing Waves A standing (or stationary) wave occurs when two waves of the same frequency pass through each other in opposite directions.

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Formation of a Standing Wave

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Waves Travelling in Opposite Directions: The diagram shows two waves (in green and blue) travelling in opposite directions on a string, setting up a standing wave (in black). Note that the standing wave is the sum of the two travelling waves.

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Energy in a Standing Wave No energy is transferred by a standing wave. Energy is trapped in the wave, changing between kinetic and potential energy as particles vibrate back and forth. Standing waves is a resonance effect, because the superposition of the two waves of identical frequency results in an increased amplitude of oscillation.

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Properties II Nodes are points on the standing wave that remain stationary at all times. Antinodes are points on the standing wave that have the greatest displacement.

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Harmonics in a String The 1 st harmonic in a string has a wavelength of twice the length of the string. The 2 nd harmonic has a wavelength equal to the length of the string. The 3 rd harmonic has a wavelength equal to 2/3 the length of the string.

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Problem 1: Telephone wires often resonate and hum in high winds. When the wind blows in a particular direction, some wires are heard to resonate at 50 Hz. The telephone poles are 25 m apart. Sketch such a telephone wire vibrating at its fundamental frequency. Calculate the wavelength of the fundamental. [50 m] Calculate the wave speed. [2500 ms -1 ] Sketch the wave pattern of the second harmonic.

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Problem 2: A guitar string is 75.0 cm long. It is made to vibrate at its fundamental frequency. What is the wavelength of the vibration? [1.50 m] The frequency of the note is 465 Hz. Calculate the speed of the wave in the guitar string. [698 m s-1] What is the frequency of the third harmonic? [1395 Hz] Calculate the wavelength of the third harmonic. [0.50 m] The string is shortened to 35.0 cm as the player’s fingers move down the fret board. Calculate the new fundamental frequency. [996 Hz]

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Demonstrations Slinky – transverse waves Guitar – identify the first, second, third, fourth harmonics by forcing nodes.

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Standing Waves Part 2: Pipes (Longitudinal Standing Waves)

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Longitudinal Standing Waves When we blow across a bottle, we set up a standing wave that is at the natural frequency of the bottle. If we add water to the bottle, we have reduced the amount of space available for the wave to be set up in. The wavelength of the fundamental is also reduced.

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Longitudinal Waves A column of air inside a pipe can vibrate – these vibrations are longitudinal: they behaves like compressions in a spring.

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Open and Closed Pipes As demonstrated in the diagram below, the closed end of a pipe acts much like the fixed point of a string – here a node forms. The open end of a pipe, however, forms an antinode.

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Problem 3 An organ pipe is closed at one end and open at the other. Sketch the wave pattern of the fundamental resonance inside the pipe. The pipe is tuned to 50.0 Hz for its fundamental note. Calculate the wavelength of this note, given that the speed of sound is 320 m s-1. [6.4 m] Calculate the length of the organ pipe. [1.6 m] Sketch the wave pattern when the third harmonic is set up in the pipe. Calculate the frequency of the third harmonic. [150 Hz]

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Problem 2 Some physics students use a speaker (loudspeaker) to resonate a pipe that is open at both ends. The pipe is 1.20 m long. Calculate the wavelength of the fundamental resonance. [2.40 m] One end of the pipe is now closed off. Does the speaker need to be changed to a higher or lower pitch in order to re-tune the pipe to the new fundamental? Explain. Assuming that the speed of sound is 320 m s -1, calculate the new resonant frequency. [67 Hz]

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Practice Complete problems 1, 3, 4, 5 Complete worksheet 3 on standing waves.

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