Applications of Applications of ResonanceResonanceHarmonics and BeatsHarmonics and Beats

BeatsBeats

Not that kindNot that kind This is due to the interference of two This is due to the interference of two

waves of similar frequencies. waves of similar frequencies. They interfere in a way that you hear They interfere in a way that you hear

alternating loudspots and softspots.alternating loudspots and softspots.

HarmonicsHarmonics We are going to look at three situations, strings, We are going to look at three situations, strings,

open ended tubes and closed ended tubes.open ended tubes and closed ended tubes. Strings can have standing wave created by Strings can have standing wave created by

plucking them or by finding the resonant plucking them or by finding the resonant frequency with a tuning fork.frequency with a tuning fork.

The ends of the strings do not vibrate, therefore The ends of the strings do not vibrate, therefore they must be….. they must be…..

NodesNodes The simplest wave vibrations is when you have an The simplest wave vibrations is when you have an

antinode at the center of the string. That would antinode at the center of the string. That would show half a wavelength. Therefore the show half a wavelength. Therefore the wavelength would be wavelength would be

/ 2L 2L

If we remember our wave If we remember our wave equationequation

V=freq x wavelengthV=freq x wavelength Therefore the f = v/wavelength Therefore the f = v/wavelength For the simplest wave fFor the simplest wave f11 = v/2L = v/2L This is called the fundamental This is called the fundamental

frequency – the lowest frequency of frequency – the lowest frequency of vibration of a standing wave.vibration of a standing wave.

V is the speed of the waves on the V is the speed of the waves on the string and not the speed of the sound string and not the speed of the sound wave.wave.

Lets look at the Lets look at the harmonic seriesharmonic series

N stands for the N stands for the harmonic numberharmonic number

N=2 the wavelength= L N=2 the wavelength= L the fthe f22=2f=2f1 1 It is called the It is called the second harmonic or 1second harmonic or 1stst overtoneovertone

N=3 wavelength=2/3L N=3 wavelength=2/3L ff33=3f=3f1 1 This is the 3This is the 3rdrd harmonic and 2harmonic and 2ndnd overtoneovertone

N=4 wavelength= 1/2L N=4 wavelength= 1/2L ff44=4f=4f11

N=5 wavelength= 2/5L N=5 wavelength= 2/5L and so onand so on

Lets see the math!!!! Yee Lets see the math!!!! Yee Haw!Haw!

ffnn= n(v/2L) where n=1,2,3 and so on= n(v/2L) where n=1,2,3 and so on Lets practice!Lets practice!

What about open ended What about open ended pipes you ask?pipes you ask?

Standing waves can be setup as a column of Standing waves can be setup as a column of air in a pipe like in an organ. air in a pipe like in an organ.

If the pipe is open at an end, an antinode If the pipe is open at an end, an antinode exists.exists.

Therefore 2 open ends have two antinodes. Therefore 2 open ends have two antinodes. This allows for all of the same harmonics This allows for all of the same harmonics

available to the string available to the string ffnn= n(v/2L) where n=1,2,3 and so on= n(v/2L) where n=1,2,3 and so on In this case v is the speed of sound in the In this case v is the speed of sound in the

pipe.pipe.

In a closed end?In a closed end?

Well… Thanks for asking… Since there is Well… Thanks for asking… Since there is that closed end, there must be a node at that closed end, there must be a node at that end. that end.

Therefore this limits the number of Therefore this limits the number of harmonics that are possible. harmonics that are possible.

The fundamental frequency consists of ¼ The fundamental frequency consists of ¼ of a wave pattern. of a wave pattern.

Therefore for closed ended only odd Therefore for closed ended only odd harmonic numbers are allowed.harmonic numbers are allowed.

ffnn= n(v/4L) where n=1,3,5= n(v/4L) where n=1,3,5