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Resonant frequencies and spectrum, a comparison between different profiles
In my previous post I mentioned the difference in timbre between a cylindrical, conical and generically shaped didgeridoo. To quantify this there is a simple and accurate mathematical construction that enables one to calculate the resonant frequencies of a didgeridoo of a given shape. It is not too difficult to implement this as a computer program and then plot the resonant frequencies to obtain a graphical representation of my qualitative statements about form and sound. Below we can see the internal profiles of four different didgeridoos. The four shapes have been chosen such that the generic didgeridoo lies between the two extremes of conical and cylindrical, while the fourth “radical” profile has been included to show the effects of more creative changes to the internal profile.
Figure 1. The profile of four different possible didgeridoos all tuned to have a fundamental tone at 60 Hz.
The acoustic impedance, plotted in the diagram below, provides information on the resonant frequencies of an instrument (corresponding to the peaks of the spectrum) and also on the backpressure (related to the relative magnitude of impedance at a given frequency). Below we see the superimposed impedance spectra of a conical, generic, cylindrical and radical didgeridoo, all tuned to have a fundamental frequency of 60 Hz.
Figure 2. The impedance spectrum showing the resonance peaks for the four different instruments of Figure 1. The vertical lines correspond to the harmonics over the common fundamental frequency of 60 Hz (at 60, 120, 180… Hz). The vertical dashed line corresponds to a note that would be a musical interval of a 10th above the fundamental frequency, in this case it is at 150 Hz.
Observing this figure we see clearly that the greatest spacing between resonances occurs for the cylindrical form and closely follows the odd harmonics of 60 Hz as one expects. The smallest spacing is for the conical profile with the second resonance being close to a musical interval of a 10th above the fundamental. The generic shape has a spectrum that lies between these two extremes while the radical profile has a less regular behaviour. This enables one to have a general feeling for the spectrum of an instrument given its internal profile. Clearly one could make strange expanding and contracting internal forms that can have greatly varied spectrums although for a completely generic form the quality of the resonances can be seriously degraded. For example one can see that for the radical profile didgeridoo the third resonance (second overtone or “toot”) at around 270 Hz has a lower impedance when compared to nearby resonances for the other three profiles. As a consequence the second toot will be more difficult to play on this instrument in comparison to the second toot on more conventional instruments.
The final figure shows a zoom on the impedance close to the fundamental frequency and here we can see that in general a conical instrument has lower backpressure than instruments with more cylindrical profiles. The radical instrument probably has a higher backpressure also as a consequence of the constriction in the first part of the profile. From figure 2. on the other hand, one sees that the conical instruments have a slightly higher back pressure on overtones than the other instruments (apart from the anomalous second toot of the radical instrument already discussed above).
Figure 3. A zoom in on the impedance spectrum of the four instruments around the fundamental frequency (at approx. 60 Hz).
One can also learn more about the actual timbre of the notes played from the impedance spectrum. The peaks also correspond to frequencies that are easier to accentuate while playing the drone and modifying the shape of the vocal cavity. The general timbre of the instrument, when playing the fundamental tone, is determined by the amount in which the various resonances of the instrument are excited by the harmonic overtone series above the drone. For example, when playing the fundamental tone on the radical instrument one should hear a strong accentuation at about 480 Hz as a consequence of the alignment between the instrument spectrum and the harmonic series at that frequency as is clear from figure 2.
An additional interesting observation is the first toot of the generic, conical and radical instruments, which is between a musical 10th and a musical 12th above the fundamental – a fact which any didgeridoo player with some experience on different instruments has surely noticed while playing.
To obtain the actual audio spectrum of the instruments, meaning the frequency components of the actual played sound, one needs to combine the above results with the vibrations of the players lips. This will be the issue of an upcoming blog post.
Gravitational Waves
The announced detection of a gravitational wave signal arriving from the inspiral of two black holes resulting in their inglobation into a final more massive black hole has now travelled around the world.
We all have an enormous practical experience in the detection of waves. Actually, almost all of the information that we receive about the surrounding world arrives to us in the form of waves: Sight – our eyes are extremely sensitive detectors of electromagnetic waves; Sound – our ears can detect a large range of sound waves; Touch – our body detects certain frequencies of electromagnetic radiation coming from the sun (it warms our skin).
Each of these waves is detected as a function of the way that it travels through an elastic medium. The medium of electromagnetic waves and also of gravitational waves is the vacuum. To observe the universe more completely science has developed detectors to extend the range of seeing and hearing well beyond the range accessible by our body, and this has enabled us to observe a huge variety of events in the universe. These detectors have extended our range of vision in electromagnetic waves well beyond the visual ight of our everyday experience – and this has led our investigation of the properties of matter and spacetime deep into the microscopic and cosmological realms. As a consequence of this extended vision the theory of quantum mechanics was developed and refined.
Gravitational waves are a simple prediction of Einstein’s general theory of relativity, a theory that celebrated its 100th anniversary in November of 2015, and which continues to be confirmed as a spectacularly successful theory of gravity with no close contenders. The construction of gravitational wave detectors began in the early 1970’s but up until last year they were never sensitive enough to detect the gravitational waves that we expected should arrive from cataclysmic events in remote regions of the universe. The construction of gravitational wave detectors is extremely demanding due to the incredible weakness of the gravitational field and it is only with the dedication of experimental physicists continually refining the detectors that we have finally arrived at the actual observation of at least one, and probably various other, gravitational waves.
The magnitude of this discovery is completely out of reach or our everyday experience. The difference in strength between the gravitational force and the electromagnetic force is on the order of forty zeroes. Forty zeroes. This number is really beyond imagination. If we take a huge number, for instance the distance to the big bang is on the order of twenty zeroes in seconds, then it is still excruciatingly small compared to a number with forty zeroes.
What will the future bring now that we have opened a new window of perception on our universe? The most evident lessons are related to black hole physics. The observation of this gravitational wave is the most direct evidence of the existence of black hole like objects almost all the way to their horizon – the famous point of no return and the source of all the subtleties of black hole physics.
Black holes, like gravitational waves, we’re first discovered as solutions to Einstein’s equations almost 100 years ago. It is beautiful to ponder that these two predictions are finally coming into view at the level of observation and are amongst the more profound confirmations of Einstein’s theory and at the same time the most likely to lead to the further evolution and extension of this theory into the quantum world.