Using Schlieren imaging to estimate the geometry of a shock wave radiated by a trumpet bell

The techniques used to describe acoustic pressure fields usually involve measurements with microphones, and although these methods work perfectly well for a wide variety of applications, they are not well-suited for visualization of acoustic waves as they are mostly incapable of portraying the geometry of the wavefronts without the microphones, of which there would typically be a large number, themselves becoming intrusive. Optical methods, such as the Schlieren and shadowgraph techniques, have been used successfully in a variety of settings in the context of both fluid dynamics and acoustics.¹ The Schlieren method, in particular, can be used to observe shock waves because the sudden rise in density which characterizes them gives rise to a steep gradient in the refractive index of air, which in turn makes the shocks detectable by this method.² Pandya et al.³ and Hargather et al.⁴ have examined numerous systems which produce weak shocks susceptible to visualization by means of the Schlieren method. Pandya et al.³ also discuss the conditions under which it is possible to observe weak shocks, concluding that very high excitation amplitudes of high-frequency signals are ideal for this purpose. One system examined in both papers is a trumpet being played at high dynamic levels. As first established by Hirschberg et al.,² shock waves can develop inside brass instruments as a result of nonlinear steepening for sufficiently intense waves produced in sufficiently long instruments. Schlieren images of shock waves radiated from a trumpet bell are shown in both aforementioned papers. No attempt was made in either paper, however, to obtain the geometry of the shock fronts directly from the images, or to calculate parameters of the propagation using these same images. The main objective of this paper is then to examine whether the Schlieren method can be used to describe quantitatively the geometry and propagation of weak shock waves with sufficient accuracy to model the radiation of waves with different frequencies from a trumpet bell, and then also to estimate the instantaneous propagation speed of these waves.

Many important properties of wind instruments depend mostly on the geometry of the instruments, such as the instrument cutoff frequency. Below this frequency most of the energy associated with outgoing waves will be reflected before the open end of the bell is reached, while above it most of the energy will be radiated by the bell. The geometry of the bell also determines, however, at which point along the instrument these waves are radiated, and thus, by extension, what the precise shape of these radiated wavefronts will be. This location also depends on the frequency of the outgoing waves, with high-frequency waves detaching from instruments with rapidly-growing flares earlier than low-frequency waves. The radiation of acoustic waves from horns of varying geometries has been the subject of a great number of studies, most of them proposing one-dimensional models where propagating wavefronts are described in terms of time-invariant equipotential surfaces where a velocity potential is constant. Because plane waves cannot meet the walls of a flaring horn at straight angles, spherical surfaces are often used to approximate the geometry of the actual wavefronts. The validity of this approximation was famously established first by Benade and Jansson^5,6 and later also by Eveno et al.⁷ Although, in principle, Schlieren images of shock waves radiated from a trumpet bell should permit a precise description of the geometry of the wavefront to be obtained, in practice, the spatial resolution of these images was not sufficient for this precise description to be feasible, and we did approximate the radiated wavefronts by means of spherical surfaces.

The experimental setup was designed to allow for the visualization of shock waves being radiated from the open end of a trumpet and for the simultaneous recording of the pressure field at a fixed distance from the exit of the trumpet bell and along the central axis of the instrument. We used a Silvertone $B♭$ trumpet, considered to be a good quality student instrument, with no valves depressed, a position corresponding to the note $B♭3$ and a frequency of 233 Hz. Sound excitation was provided by means of a Radson (Guadalupe, Mexico) U150 S driver, coupled to the mouthpiece-end of the instrument—without a mouthpiece—by means of a custom-made PVC cylindrical tube with a length of 8.4 cm and an internal diameter of 10.7 mm. In order to produce shock waves inside the trumpet resonator, very high excitation amplitudes are required, as shown by Rendón et al.⁸ A Stanford Research (Sunnyvale, CA) DS345 function generator and a Yamaha (Shizuoka, Japan) AX-380 power amplifier were used to feed high-amplitude trains of sinusoidal signals with durations of 10 ms into the driver. Short trains were used instead of continuous signals to preserve the integrity of the driver membrane, which can be damaged easily at sufficiently high amplitudes when responding to single-frequency signals. The pressure field was measured at a distance of 6 cm from the open end of the bell along the central axis of the instrument using a Brüel & Kjaer (Naerum, Denmark) type 4165 probe microphone setup along the instrument axis so as to minimize interference with the radiated pressure field. These measurements were then acquired and transferred to a laptop computer using a Brüel & Kjaer type 2034 dual-channel analyzer and a National Instruments (Austin, TX) GBIP/IEEE-488.1 interface, respectively.

Visualization of the radiated shock waves was achieved through a traditional single-mirror Toepler Schlieren system, depicted schematically in Fig. 1. The optics of the system consist of a spherical mirror with a diameter of 15 cm and a focal distance of 1.5 m, a beamsplitter, and a couple of focusing lenses, with focal distances of 18 cm, next to the camera, and 3 cm next to the light source. As a light source we used a 532 nm, 2 W laser made by Laser Quantum (Cheshire, UK), and as a recording device we used a Miro M310 Phantom camera (Vision Research, Wayne, NJ), capturing images with a resolution of 256 × 128 pixels at a rate of 64 000 frames per second, with an exposure time of 1 μs. The trumpet was placed so that the open end of the bell was barely visible at the edge of the mirror.

Pandya et al.³ report that the Schlieren method can be used to visualize shock waves produced by notes above concert F5, or a corresponding frequency of 700 Hz, played at fortissimo levels. Based on these recommendations, we modulated single-frequency signals corresponding to 700, 1400, and 2100 Hz in order to produce short wave trains at these frequencies. Notice that F5 is the third harmonic of $B♭3$⁠, so that excitation of these frequencies within the instrument would give rise to resonant modes, resulting in maximum oscillation amplitudes. Note also that the lowest frequency falls below the instrument cutoff frequency, while the highest frequency falls above this frequency—Benade⁹ estimated the cutoff frequency for a trumpet to be between 1300 and 1600 Hz. In the event of shock formation, however, energy is pumped to the higher end of the frequency spectrum, prompting increased radiation from the trumpet bell. In order to achieve shock formation inside the instrument, the excitation amplitude was in all cases adjusted so that the root-mean-square value of the pressure calculated for each of these wave trains, as recorded by the probe microphone, was at least 130 dB. Both the visualization of a shock wave produced at 700 Hz and the corresponding time-signal recorded by the microphone are shown in Fig. 2. The fundamental frequency of the train of shock waves radiated from the open end of the bell coincides with the frequency with which the waves were generated by the sound source.

Given the limited resolution of the obtained images, seen in Fig. 2(a), any measurement of the shock geometry will have a non-negligible degree of error. In order to capture the geometry of the shock wavefronts we chose to fit circles to the two-dimensional (2D) visualized curves using a standard least squares algorithm. The main reason for choosing to fit a circle rather than an arbitrary curve to the available data points is that it is essentially impossible to calculate the radius of curvature from a digital 2D image with good accuracy, mainly because this parameter is inversely proportional to the second derivative of the curve in question, and the estimation of the higher derivatives of functions with values specified with limited precision is very problematic.¹⁰ In addition, it has been shown experimentally that for most purposes considering a spherical shock profile is a good approximation for the geometry of a wavefront radiated from an instrument bell,^6,7 although there is also compelling evidence that this approximation is far from perfect.^11–13 Thus, in order to estimate the radii of curvature of the shock fronts along the central instrument axis, we first located the shock region in a particular frame and placed a large amount of points on the left boundary of this region. The position of these points was saved, the points were removed, and then new points were placed over the center of the region. In a final iteration of the procedure, points were placed on the right boundary of the region. This procedure is illustrated in the inset in Fig. 2(a). We then proceeded to fit a circle to the set of all points generated in this manner, making the calculation of both the radius of curvature and the center of the circle a straightforward matter.

As we have discussed in Sec. 1, we expect higher-frequency wave trains to detach from the instrument earlier than lower-frequency wave trains. Since the slope of the horn profile is monotonically increasing, this implies that higher-frequency wavefronts should have larger radii of curvature than lower frequency wavefronts as they separate from the trumpet bell and at all distances thereafter. This behavior can be clearly seen in Fig. 3(a). We also observe that for all three frequencies the radius of curvature increases linearly with distance from the bell, which would indicate that the propagation was indeed spherical if the spheres all shared the same center. The slopes of the straight lines corresponding to frequencies of 700, 1400, and 2100 Hz are 0.928 ± 0.017, 0.867 ± 0.038, and 0.833 ± 0.042, respectively, where uncertainties represent the standard error of the estimated slopes. Note that these results are in line with those reported by López-Carromero et al.,¹⁴ where it is assumed that the source of radiation is a virtual monopole located inside the instrument. In this case, the propagation distance is equal to the radius of the spherical wavefront. However, as we see in Fig. 3(b), the position of the virtual source does not stay put and does move slightly toward the open end of the bell. In consequence, the profile is not entirely spherical. Even when the slope in this case is obtained from data with error which is not small, a simple statistical analysis gives a good p value, lower than 0.001 in all cases, indicating that this tendency is not likely to be a consequence of the error.^15,16 These results are again consistent with those described by López-Carromero et al.,¹⁴ where it is also noted that the small movement of the virtual source is due to the rapid flaring of the instrument bell. Furthermore, this plot also shows that virtual sources are positioned further away from the open end of the bell with increasing frequency, as expected.

To confirm the validity of our measurements so far, we calculated the sound speed of the propagating shock fronts. Successive frames were used to estimate the instantaneous velocity of the radiated wavefronts along the central axis of the instrument. The results are plotted in Fig. 4, where it is shown that, as anticipated, the propagation velocity does not seem to depend on frequency. The propagation speed just outside the bell is barely supersonic, and it decreases with distance from the bell, tending toward the local sound speed due to spherical spreading of the propagating wave.

We have confirmed by means of Schlieren visualization that radiation of weak shock waves from the bell of a trumpet is practically spherical, as already suggested by a number of authors.^5,6,7,14 By taking into account three different frequencies which corresponded to harmonics of the fundamental frequency associated with the instrument with no valves depressed, it was possible to obtain amplitudes high enough to guarantee the generation and eventual radiation of weak shock waves. Pandya et al.³ had already suggested that high-amplitude single-frequency waves give rise to shock waves with the same frequency, and our results are in line with their findings. It was then possible to observe that the location at which the shock waves detach from the instrument depends on their frequency, and that the increased directivity of high-frequency waves is consistent with our results. The Schlieren method cannot only be used to quantify geometric details of the wavefronts shown on stills, but can also provide a good estimate of the speed of propagation of the radiated shock waves based on the analysis of a sequence of stills. Based on these results, we believe the Schlieren method can be a useful tool not only for the visualization of an assortment of acoustical phenomena, but also for the quantitative analysis of said phenomena.

The authors gratefully acknowledge financial support granted by DGAPA-UNAM through Project No. PAPIIT IG100717. We also acknowledge the help given us, always cheerfully, by Tomás Vielle.