For the final instalment of this chapter on measurements, we will give a brief summary of some ways to see sound fields and other air-flow phenomena with musical connections. There are no easy ways to do this: all the methods to be discussed involve laboratory-grade hardware, and most of them involve serious computing power as well.
We will start with ways to visualise acoustic fields. The first is the simplest: if we have a steady source of sound, we can measure the sound with microphones at a large number of places (keeping careful track of the relative phase of all the recordings). We can then simply plot the sound pressure at all the points. In practice, we probably don’t measure every position with a separate microphone: we can use a small number of microphones, or even just a single microphone, and move them around to cover the desired measurement grid.
Figure 1 shows an example. A trombone was driven sinusoidally, by a loudspeaker at the mouthpiece end, in an anechoic chamber to eliminate wall reflections. A vertical line of 23 microphones was used to record the sound at different distances away from the trombone bell. The lowest microphone in the line was always on the axis of the trombone bell, while the others were spaced out above it. These signals, phase-coordinated by the sinusoidal drive signal, were then combined to produce the plots.
Four different frequencies are shown: lower ones in the top row, higher ones in the bottom row. At the lowest frequency, the contour lines of pressure in the spreading sound wave are almost quarter-circles: the wavelength is a lot longer than the diameter of the bell, and the sound field is approximately omnidirectional. As the frequency goes up, the sound is concentrated progressively into a “beam” along the axis. At the same time you can see the wavelength getting shorter.
There is another, more sophisticated, way to make use of information from a grid of microphone measurements, called nearfield acoustic holography, or “NAH”. We are more familiar with holography applied to light waves and visual images, but sound waves obey the same mathematical wave equation and the approach carries over to acoustics. Indeed, it works better with acoustics.
The reason is the very high frequency of light waves, compared to sound waves. It is not feasible to record light signals in enough detail to preserve the phase information. In optical holography, this problem is overcome by mixing light from the test object with a reference beam: the interference pattern conveys some phase information. But in acoustic holography, it is perfectly possible to record full information from all the microphone positions.
A typical test setup for NAH is sketched in Fig. 2. There is a rectangular grid of microphone positions positioned parallel to the top plate of a guitar, a very short distance above it. Signals are recorded at all these positions when the guitar is excited. This excitation could be with an impulse hammer, a shaker, or through the strings: anything that is accurately repeatable, or measurable so that each signal can be turned into a frequency response function.
These signals are then processed in a way that is a kind of mirror image of something we have met before. Back in section 4.3.2 we looked at the way sound is radiated from a vibrating plate. Provided the plate is flat, and surrounded by a rigid baffle, the radiated sound at any position can be calculated. Each small piece of the vibrating plate behaves like a monopole sound source, and we know exactly how the resulting sound field behaves (see section 4.3.1). All we need to do is add up these monopole contributions from each little piece of the plate, in the form of an integral called the Rayleigh integral.
So in summary, if the motion is known at every point in a plane, we can compute the resulting sound field at any position in space. NAH takes advantage of a corresponding property: provided the sound pressure waveform is known at every point in a plane, there is an integral formula to reconstruct the sound field at other positions, and also to reconstruct the motion of the sound source, in our case the vibration of the guitar top plate. The mathematical details are a little more complicated than in the case of the Rayleigh integral: see the next link for a brief description. For the method to work at its best, the measurement plane has to be very close to the source plane: this is where the “nearfield” gets into the name, as explained in the link.
NAH was developed around 1980, and one of the early applications was indeed to guitar vibration using a measurement setup very much like Fig. 2 [4]. We can show some results from this early study. A $16 \times 16$ grid of microphones was used, and the guitar was excited by driving one of the strings electromagnetically. The string’s tuning was then adjusted to give results at different frequencies. Figure 3 shows results for the reconstructed velocity over the guitar top, at a range of frequencies. One thing is immediately clear: this method of reconstructing the “plate vibration” also gives a direct visualisation of the motion of the “Helmholtz piston” in the soundhole.
At the lowest frequencies, the images are indeed dominated by the soundhole piston. This is to be expected: the “air resonance” of a typical guitar is around 100 Hz, so this mode will dominate for all frequencies in the top row of the figure. As frequency rises we see more motion in the top plate, but it is still accompanied by significant soundhole motion over the entire frequency range explored here.
These results link to something we saw in section 10.5. Figures 14, 17, 20 and 23 in that section showed animations of modes of a violin body, which included a representation of the airflow through the f-holes. Those results, due to George Bissinger, were obtained by combining his modal tests of the mechanical parts of the violin body with NAH results from a measurement array just covering the region of the f-holes.
Figure 4, from the same guitar study [2], illustrates another aspect of NAH reconstruction. The chosen frequency here corresponds to a strong modal response of the guitar, described by the authors as a dipole mode. It probably has a mode shape somewhat similar to the lower left plot of Fig. 4 in section 5.3, but that figure related to a classical guitar whereas the present results are for a steel-string guitar, so the correspondence of modes may not be exact. Figure 4 shows the sound field, reconstructed at several distances away from the guitar top. The arrows show the magnitude and direction of acoustic energy flow at each point.
The upper plot shows results on a plane perpendicular to the guitar top and aligned with the long axis of the instrument. The lower plot shows a section on a transverse plane through the bridge. The upper plot shows sound energy mainly originating at the soundhole, and radiating radially outwards. The lower plot shows something less obvious: the dipole nature of the top-plate motion results in sound energy coming out of the left half of the top, and some of it looping round and going back into the top plate on the right-hand side. This a typical near-field effect: local motion, not leading to far-field sound radiation.
The other NAH example we will show comes from the work of Lily Wang and Courtney Burroughs [3]. They surrounded the body of a violin with four planes of holographic measurements, each consisting of a $120 \times 120$ grid of microphone positions. These results allowed them to capture the full three-dimensional sound field around the violin. The instrument was bowed by a mechanical bowing machine with an endless loop of horsehair, providing steady excitation for as long as the measurement sequence required.
Results were analysed separately for each harmonic of the bowed-string sound. The experimental technique produces a very large amount of data, and the full three-dimensional sound field is hard to visualise, so we will look at some sample plane cross-sections through the field. Figure 5 shows some results at the fundamental frequency of the open D string. Each plot shows sound intensity vectors, reconstructed on a plane not quite touching the surface of the violin. For the left-hand plot this plane is near the top plate, for the right-hand plot it is near the back plate.
The frequency is close to the air resonance A0, so we expect the sound radiation to be dominated by breathing of the “f-hole pistons”. In these near-field reconstructions it can indeed be seen that the strongest sound intensity comes from the central region of the top plate. In the back view, there is little sound coming direct from the plate, but we see some sound “leaking round the corner” from the front, especially in the area of the c-bouts. In the far field, we already know that the sound radiation pattern must become essentially omnidirectional because the violin body is much smaller than the wavelength of sound at this frequency.
The next pair of images relate to the sound field at the second harmonic of the open D string. Figure 6 shows a reconstruction corresponding to the left-hand image of Fig. 5, on a plane not quite touching the top plate. Figure 7 shows a different section through the same sound field, on a perpendicular plane aligned on the long axis of the violin.
Finally, Fig. 8 shows the sound field at a higher frequency, the third harmonic of the open A string. The figure shows the same section as in Fig. 7, and it can be seen immediately that the sound field is much more directional. We are only seeing that field fairly close to the violin, but at this frequency of 1320 Hz the violin body is no longer small compared to the wavelength of sound, so that we can expect the far field also to exhibit some directionality.
Now we turn to methods for visualising fluid flows, to see things that will be important in the next chapter when we look in detail at wind instruments. The oldest approach is called schlieren photography, and it pre-dates the computer era. It is an optical technique that relies on the fact that if the density of air changes, then its refractive index also changes. This is a familiar effect: one way to change the density of air is to heat it, and the optical effects of heated air give rise to shimmering heat haze, and to desert mirages. But sound waves also modify the density of air, and schlieren photography can be used to visualise some acoustical phenomena. It uses an ingenious optical setup to make the associated changes in refractive index visible as images.
Figure 9 shows one setup for schlieren photography. A light source shines on a mirror, which turns the spreading ray pattern into a parallel beam. In the absence of any disturbance, these rays (the blue lines) would then hit another mirror and be focused back down to a point. A knife-edge is placed at this point. Now the test object is inserted into the parallel beam, at the same distance from the second mirror as the knife edge. This object may deflect some of the rays, indicated by the red lines.
These deflected rays may or may not be able to get past the knife edge — the red dotted lines show rays that are blocked. The red rays are then collected by a camera lens and focused on the film or sensor. The brightness of the resulting image will be modulated, according to what proportion of the rays are able to get past the knife edge, leading to a grey-scale image that reveals the density variations caused by the test object. In a variation of the technique, the knife edge may be replaced by a filter with graduated colour, leading to a coloured image. See this wiki page for more detail about the variety of schlieren techniques, and for more images.
It is useful to see a couple of non-musical images to illustrate the method in action. Figure 10 shows a coloured image of the heat rising above a hot soldering iron. Figure 11 takes us a step closer to our real purpose, because it is showing density variations caused by pressure changes in the air. This is a schlieren image of a speeding bullet. Because the bullet’s speed is supersonic, shock waves are generated at the leading and trailing edges, giving the obvious V-shaped patterns. Shock waves like this are the cause of the “sonic boom” you hear when a supersonic aircraft flies past you. They are also the cause of the “crack” of a whip: when someone cracks a whip, they launch a wave or loop that travels down the whip, and it speeds up until some part of the whip moves at supersonic speed and generates a shock wave.
Figure 12 shows a music-related example of a shock wave, made visible by schlieren imaging. A trumpet is being driven, loudly, by a loudspeaker at the mouthpiece end. As we will see in section 11.5, a sufficiently large-amplitude wave propagating down the long tube of the trumpet can “sharpen up” and generate a shock wave. This effect is responsible for the characteristic “brassy” sound of a trumpet or trombone when played loudly. The end of the trumpet bell is visible at the extreme left of the images, and a shock wave can be seen emerging and travelling off to the right in the successive images of the set.
Figure 13 shows an example of a different kind of music-related air flow. As we will see in section 11.8, the physical mechanism of sound production in instruments like the flute or recorder involves an interaction between the acoustic pressure inside the pipe, and an air jet impinging on a wedge with a sharp edge. The dynamic behaviour of such an air jet is complicated. The jet switches between the two sides of the wedge during each cycle of the oscillation, and in the process it usually sheds vortices. Both aspects of the behaviour can be seen in this sequence of schlieren images.
Another traditional method of visualising flow patterns involves “seeding” the flow with very light particles. In a photograph with a longish exposure, the particles produce visible trails that mark out the streamlines of the flow. A modern upgrading of this approach is called particle image velocimetry, or PIV for short. If two short-exposure images are captured, separated by a very short time interval, then it is possible to process the images to reveal the magnitude and direction of the flow velocity vector at each point.
If the particles are sparse, this can be done by tracking individual particles. More commonly, a higher density of particles is used, and the computer looks for the velocity which maximises the correlation between the two images over a small region. This is repeated over a grid of small regions, giving a map of the flow velocity. Figure 14 shows an example. The main image shows a measurement of flow at the end of a vertical pipe, driven sinusoidally by a loudspeaker. The position of the pipe exit and the top part of the pipe wall can be seen at the bottom of the plot.
The grid of arrows show the deduced pattern of velocity. This reveals the shedding of vortices at this particular moment in the cycle of the sinusoidal response. The two smaller images show different aspects of this flow field: the left-hand image shows the streamlines (in black) and the flow speed (in colours); the right-hand image shows the vorticity of this flow. We will explain vorticity properly in section 11.2.1, but roughly speaking the colour scale encodes the local rotation implied by the measured velocity field: clockwise in blue, anticlockwise in red.
[1] J. Kemp, A. López-Carromero and D. M. Campbell, “Pressure fields in the vicinity of brass instrument bells measured using a two dimensional grid array and comparison with multimodal models”, Proceedings of the 24th International Congress on Sound and Vibration, London (2017).
[2] W. Y. Strong, T. B. Beyer, D. J. Bowen, E. G. Williams and J. D. Maynard, “Studying a guitar’s radiation properties with nearfield holography”, Journal of Guitar Acoustics 6, 51—59 (1982).
[3] Lily M. Wang and Courtney B. Burroughs, “Acoustic radiation from bowed violins”, Journal of the Acoustical Society of America 110, 543—555 (2001).
[4] A. López-Carromero, D. M. Campbell, J. Kemp and P.L. Rendon, “Validation of brass wind instrument radiation models in relation to their physical accuracy using an optical schlieren imaging setup”, Proceedings of Meetings in Acoustics, 28, 035003 (2016).