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Applications of Resonance Harmonics and Beats

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Beats Not that kind Not that kind This is due to the interference of two waves of similar frequencies. This is due to the interference of two waves of similar frequencies. They interfere in a way that you hear alternating loudspots and softspots. They interfere in a way that you hear alternating loudspots and softspots.

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Harmonics We are going to look at three situations, strings, open ended tubes and closed ended tubes. We are going to look at three situations, strings, open ended tubes and closed ended tubes. Strings can have standing wave created by plucking them or by finding the resonant frequency with a tuning fork. Strings can have standing wave created by plucking them or by finding the resonant frequency with a tuning fork. The ends of the strings do not vibrate, therefore they must be….. The ends of the strings do not vibrate, therefore they must be….. Nodes Nodes The simplest wave vibrations is when you have an antinode at the center of the string. That would show half a wavelength. Therefore the wavelength would be The simplest wave vibrations is when you have an antinode at the center of the string. That would show half a wavelength. Therefore the wavelength would be

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If we remember our wave equation V=freq x wavelength V=freq x wavelength Therefore the f = v/wavelength Therefore the f = v/wavelength For the simplest wave f 1 = v/2L For the simplest wave f 1 = v/2L This is called the fundamental frequency – the lowest frequency of vibration of a standing wave. This is called the fundamental frequency – the lowest frequency of vibration of a standing wave. V is the speed of the waves on the string and not the speed of the sound wave. V is the speed of the waves on the string and not the speed of the sound wave.

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Lets look at the harmonic series N stands for the harmonic number N=2 the wavelength= L the f 2 =2f 1 It is called the second harmonic or 1 st overtone N=3 wavelength=2/3L f 3 =3f 1 This is the 3 rd harmonic and 2 nd overtone N=4 wavelength= 1/2L f 4 =4f 1 N=5 wavelength= 2/5L and so on

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Lets see the math!!!! Yee Haw! f n = n(v/2L) where n=1,2,3 and so on f n = n(v/2L) where n=1,2,3 and so on Lets practice! Lets practice!

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What about open ended pipes you ask? Standing waves can be setup as a column of air in a pipe like in an organ. Standing waves can be setup as a column of air in a pipe like in an organ. If the pipe is open at an end, an antinode exists. If the pipe is open at an end, an antinode exists. Therefore 2 open ends have two antinodes. Therefore 2 open ends have two antinodes. This allows for all of the same harmonics available to the string This allows for all of the same harmonics available to the string f n = n(v/2L) where n=1,2,3 and so on f n = n(v/2L) where n=1,2,3 and so on In this case v is the speed of sound in the pipe. In this case v is the speed of sound in the pipe.

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In a closed end? Well… Thanks for asking… Since there is that closed end, there must be a node at that end. Well… Thanks for asking… Since there is that closed end, there must be a node at that end. Therefore this limits the number of harmonics that are possible. Therefore this limits the number of harmonics that are possible. The fundamental frequency consists of ¼ of a wave pattern. The fundamental frequency consists of ¼ of a wave pattern. Therefore for closed ended only odd harmonic numbers are allowed. Therefore for closed ended only odd harmonic numbers are allowed. f n = n(v/4L) where n=1,3,5 f n = n(v/4L) where n=1,3,5

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