Is There an Exponential Relationship in Acoustic Resonance Amplitude Decay?

[Elle90]

Set-up
Closed tube with a hole at each end. Microphone at one end attached to computer. Clap noise made at the other end which will produce a range of frequencies. The computer program detects the resonant frequencies and their amplitudes.

The graph produced seems to show the lower resonant frequencies to have higher amplitudes and I've read that in musical instruments each new 'generation' of harmonic (increasing node number) has a lower amplitude.

On my graph it looks like the amplitude exponentially decays with increasing resonant frequency. Is this correct? Is there an exponential relation that can be seen from wave equations, if so which equation?

 

[Stonebridge]

The harmonics present, and their relative strength/amplitude, is rather complex to analyse.
In musical instruments, the harmonics present and their relative amplitudes depend on a number of factors. Probably the most important being
-the method of excitation of the air
-the shape of the tube
In general, certain harmonics can be favoured by these factors, and the pattern you mention is not "exponential"; though as a general rule, the fundamental is the greatest and higher harmonics have smaller amplitudes.
In your experiment, the diameter of the tube will influence the relative amplitudes of the higher harmonics. This is due to the "end correction"; the fact that the effective length of the pipe depends on its width.
The harmonics present also depend on whether or not the vibrations are forced or natural.
I did a search for something more rigorous but didn't find anything much.
I would suggest in your experimentation that you investigate the effect of using pipes of different width. Or, more accurately, different width to length ratios. You should notice that the relative frequencies of the higher harmonics change.
There is a link here to an article on the voicing/tone quality of organ pipes which goes into this in some detail. You will need to plough through a lot of specialised stuff on organ pipes, but the section (linked) is quite informative.
Remember that organ pipe vibrations are forced.
http://www.pykett.org.uk/how_the_flue_pipe_speaks.htm#Timbre

 

[astrokreng]

Thank you for sharing your acoustic resonance experiment set-up and results. It is interesting to see how different frequencies and amplitudes can be detected and measured using a computer program.

To answer your question, yes, it is possible for an exponential decay to be observed in acoustic resonance experiments. This phenomenon can be explained by the wave equation, which is a mathematical representation of how sound waves propagate through a medium.

In the case of a closed tube with holes at each end, the sound waves produced by the clap noise will travel through the tube and reflect off the closed end. This creates standing waves, where the sound waves interfere with each other, resulting in nodes (points of zero amplitude) and antinodes (points of maximum amplitude).

The wave equation that describes this phenomenon is the Helmholtz equation, which takes into account the properties of the medium and the boundary conditions of the tube. When solving this equation, it is possible to observe an exponential decay in the amplitude of the standing waves as the frequency increases.

This exponential relationship can also be seen in other types of wave equations, such as the damped wave equation, which takes into account the dissipation of energy in the medium. In this case, the amplitude of the standing wave will decrease exponentially over time.

In conclusion, your observation of an exponential decay in the amplitude of resonant frequencies is correct and can be explained by the wave equation. Further experiments and analysis can help to better understand the relationship between frequency and amplitude in acoustic resonance.