Impulse and Power for Complex Systems
Supplement for chapter 12
Is a tuning fork complex? Yes!
The tuning fork is famous for vibrating as described in Why You Hear What You Hear, the tines moving alternately away from each other and toward each other. But like any fairly stiff metallic object, there are hundreds and thousands of other modes.
Prof. Daniel A. Russell of Penn State University has made very compelling moving illustrations of some of these modes. We show two of these here, with permission. But you really should go to this link, that shows many more modes and has much more information.
The link shows that in spite of the tuning fork's role as a producer of a single tone, i.e. the lowest mode, shown on the left, (426 Hz in this case), other modes are very likely excited depending on how the fork is struck. See the below animated example, also appearing in "Exciting modes by the way you strike" in the web page for chapter 8.
Project: Excite a tuning fork
A tuning fork is fairly easily procured. Try high school bands. They are very inexpensive on the internet.
This project explores some of the issues just raised. The idea is to experiment with different ways of hitting the fork, and then measuring the sound carefully, preferably very close to the fork to avoid short circuiting, but being careful to avoid the silent nodes (i.e. move the mic around). Use a small USB dictation or gaming mic. One objective is to see what modes can be excited, or at least get their frequencies, and their Q's. The Q's will differ a lot since some modes will shake the handle that you are holding onto; your fleshy hand will quickly damp those modes.
Try hard hammers and softer ones, and vary the striking place.
Project: 30 degree wedge with a source
We spend considerable time with the 30-degree wedge billiard in Why You Hear What You Hear (including chapter 7 and chapter 12). Most of the illustrations were. This zip file is a saved state of Harvard Ripple for the wedge. Click on it to download and unzip it. Using the form on this link, choose the unzipped file from the place it exists on your hard drive. Once chosen, click on Submit and the "job" should come up on screen running Ripple after a short wait.
The source in the wedge as the simulation opens is operating at a frequency corresponding to a minimum of power radiation, although there is a transient period as it starts up. It soon settles down to one of the states as seen in Why You Hear What You Hear.
One way to understand how so little radiation can be emitted at this frequency is to recognize that there are two sources emitting sound down the wedge. One is the source itself; part of its emission heads straight out the wedge. But the same source emits waves which head inward and collide with the apex of the cone; these then emerge propagating outward also, but they are phase shifted compared to the direct outward emission from the source. If this phase shift is 180 degrees, the two sources destructively interfere with each other, and little emission takes place.
If you change to any other frequency, wait for things to settle down, the phase will have changed, the destructive interference will be incomplete, and more power will be emitted.
Try moving the source off center. If you move it along the center, you are in effect changing the wavelength, by the similarity principle. For example, you could move the source out by a factor of 2, and lower its frequency by the same factor, and recover the same result, magnified by a factor of 2.
"Soundboard" files
The two sounds mentioned in this chapter show the effect of overlapping resonances versus nonoverlapping resonances. When these sound files are spectrally analyzed, the nature of the overlap is clear (see Chapter 12). These are not recorded soundboards, but rather files I concocted in Mathematica to show the effects of overlap. The nonoverlapping case is irksome, and if soundboards were like that, well, we wouldn't like it.
Modeling a soundboard
The impulse response for a model of a soundboard (unspecified shape and strike point, but typical) is shown below at the bottom, and the spectrum is given at the top,
The full research paper may be found *** Link to pdf does not work *** here. The researchers are modeling a complete piano from physical principles, and we focus here on the soundboard. Note that although there are some rather tall peaks around 1000 Hz, they are still overlapping, and well into the Schreoder regime.
Schroeder frequency, impulse response
We distinguish complex vibrating systems from "simple" one dimensional objects like a stretched string, or a narrow pipe with air in it, or a Helmholtz resonator in its lowest, Helmholtz mode.
Already in two dimensions, or the three dimensions describing air in a room, or the vibrations of a soundboard, the motion, modes, and spectra take on more complex behavior. An object made of many parts and facets, like a violin body, is even more complex.
We need to understand such systems because they are often the ultimate source of sound. A complex sounding board or violin body ultimately radiates the energy originally contained in a "simple" one dimensional string. A complex concert hall receives the sound from a flute emitting fairly pure notes from a spot on stage. What then does it sound like in the seating area if the hall is full of resonances?
Manfred Schroeder (1926-2009) made a lot of progress with questions like this. He had a huge influence on acoustics. He was also a gifted mathematician. He worked for many years at Bell Laboratories. When the author visited him in his Gottingen University office in 2004, he asked me how I found the acoustics there. I thought it was excellent with a good balance between too dead and too live. He asked me to look around, and all I saw was beautiful hardwood paneling affixed to brick or forming cabinetry. There was no sound damping anywhere it seemed. So why wasn't the space way too reverberant?
Schroeder revealed the secret to me: all the panels were affixed in the back to a dense collection of damped springs anchored to the brick wall or hardwood board behind. The springs were resonant over a range of frequencies, so that the panels had a much higher absorption of sound at most frequencies than they looked like they could. He was very proud of this design, but one of hundreds of acoustical innovations developed over his lifetime.
Why You Hear What You Hear spends considerable time explaining and illustrating the concept of the Schroeder frequency, above which the spectrum of room resonances go from isolated to overlapping; i.e., the resonance widths start to become considerably larger than the spacing of the frequencies of the adjacent resonances. There are all sorts of implications of this. The power spectrum does not cease to oscillate above the Schroeder frequency, but rather the oscillations take on a characteristic spacing and "contrast" (height difference between peak and trough of the oscillations) which have been well-characterized by Schroeder.
The peaks and valleys in the region above the Schroeder frequency are collections of many individual modes, but the modes themselves are becoming a weak foundation for understanding because they are so short lived. A much better place to hang your hat is in the time domain, where the bumps and dips in the autocorrelation become predictable as echoes arriving at specific times from walls and other objects within the T60 reverberation time. This is typically many times shorter, and perhaps thousands of time shorter, than the time required to distinguish one mode from another (even though that isn't even possible in principle because they overlap badly) above the Schroeder frequency. This is an uncertainty principle issue: if two modes differ by delta f in frequency, it takes a time of order of 1/(delta f) to distinguish them. In the mid to upper ranges, this could be thousands of seconds for a moderate sized room.
All this argues for an impulse response approach to room acoustics, this is already standard, an moreover and impulse approach to many other vibrating objects, like piano soundboards and cello bodies, where it is much less common.
Schroeder frequency for concert halls vs. objects.
Although both concert halls and complex objects like a violin body have overlapping resonances above a certain frequency, there can still be considerable "structure" in the form of peaks involving many modes collectively. Depending on shape, concert halls impose certain limits on the strength of these peaks, as discussed in Why You Hear What You Hear. Objects, depending on how they are constructed, can have quite strong peaks that definitely affect their timbre, for better or worse.
Here, we see impulse response and spectrum of a violin body, as measured from the side of the bridge. The study is available here on the web.
The peaks to valleys here in the response spectrum above the Schroeder frequency ( a little hard to say where this is without a modal analysis) often registers 30 dB, whereas it is typically 11 dB in a concert hall.
The Schroeder Frequency and the Sabine T60
This is an "engineer's formula"; the factor or 2000 is not actually dimensionless, but it works if V is given in terms of cubic meters and T_60 of course is in seconds, then f_s is the Schroeder frequency in Hz.