Our results reveal highly variable source excitations from event-to-event (Fig. 2) and systematic frequency partitioning for individual events (Fig. 3c, d). These observations have implications for the interpretation of source processes that occur in a wide range of observational settings including the interpretation of repetitive sources found in both natural acoustic and seismic data (e.g., Danesi et al. 2007; Park et al. 2019).
Our experimental setup produces a source that has conceptual similarity with prior work in volcano acoustics. For example, Buckingham and Garces (1996) presented a canonical model of volcano acoustics, providing an analytic solution for the upgoing sound field from a resonant magma (or gas-filled) conduit, with the trigger-mechanism pressure excitation function provided by a bubble pulse airgun-like source signature at depth in the conduit. We also have a submerged bubble expansion source, but in our case the wavelength of the bubble oscillations presumably approach or exceed those of the barrel container, and the bubble expansion is shallow enough that the full bubble pulse oscillations are not completed before the material is ejected. Thus, our source is more complicated than a deeper more pressure confined air-gun-like source (Buckingham and Garces 1996). Ichihara et al. (2009) discusses a related case of lake water surface explosions (not confined by conduit walls). In addition, our source is conceptually similar to the directed blast numerical simulations of Watson et al. (2021). We note that Watson et al. (2021) observed enhanced high frequencies in the downstream blast direction (i.e., above the vertically directed source), consistent with our observations.
Variation of a repeating source
While we attempted to produce identical acoustic impulses event-to-event, strong variations in waveforms and spectra for the ten events testify to an unstable source process. This is remarkable, considering that the source location was stationary relative to the sensor array and our methodology attempted to produce a uniform source discharge. Even repeat discharges with the same cannon orientation (Experiment 11 and 12) produced somewhat different waveform and spectral characteristics (see Figs. 2, 3, 5).
In natural systems, the observed waveform is interpreted to be dominantly impacted by four elements: the source, path, site and instrument. If the source mechanism is identical for two or more events at a stationary position, then the resulting waveforms would be highly similar because each of the four elements would be nearly constant over short inter-event times (e.g., Green and Neuberg 2006; Park et al. 2019) related to a constant eruption trigger, barrel geometry, water fill level and barrel opening producing highly similar waveforms and spectra. In this context, if we return to the multi-peaked spectra for events 1, 2 and 6 (Figs. 4, 5), the observed spectral peaks might hypothetically be related to a path effect associated with a Lloyd mirror (e.g., Carey 2009) where direct and ground reflected waves might be superimposed to produce constructive and destructive wavelets dependent on the distance from the source. In our case, however, the peaks persist for all stations consistent with a source rather than path effect. Instead, the systematic similarity of all spectra for a given experiment (Figs. 4a, c, 5a) suggest that the source is dominating the spectra and that some features experiment-to-experiment are unconstrained and highly variable.
We suggest that the non-systematic features of the experiment-to-experiment source process may relate to non-uniform strength characteristics of each plastic bottle as well as small variations in the position of the bottle at the base of the barrel. We also surmise that rapid bottle rupture would displace the bottle laterally along the base of the barrel in a somewhat random manner. The bottle rupture process may introduce unconstrained energy directivity effects. In addition, while we attempted to control the capacity of liquid nitrogen (N2) into each bottle, prior experiments (e.g., Wadsworth et al. 2018 and other unpublished experiments including a subset of our team) showed that larger volumes of N2 increased the explosion yield. We estimate errors in N2 bottle fill and water barrel fill to be within 5%, while the barrel inclination is known to within a couple of degrees. Finally, we note that the inclination of the barrel would produce distortions in the water height relative to the source depth. Together, these small variations probably contributed to the non-repeating rupture-time histories and attendant variation in the spectral characteristics.
In addition to the highly variable source-time histories, careful inspection of Fig. 2 suggest that waveforms may also be systematically evolving through time. We consider two aspects of the experiment that might contribute to the evolution of the waveform features. First, the bottle type was changed after the sixth experiment when the supply of L&P type bottles was exhausted. Later explosions had generally greater overpressures (Table 2) and we surmise that the MY bottles used for experiments 6–12 failed at higher internal pressure thereby initiating the expulsion event with gas at a higher pressure. Second, over the progression of the experiment, the plywood floor of the trailer began to fracture and deform. By the end of the experiment, the floor rested on the steel under-structure which was also progressively deformed (Additional file 2: Fig. S1). We regard this slow destruction of the cannon’s trailer frame as an analog to a destructive source process in nature although the temporal variability is difficult to constrain in our case. For the natural system, this might be equivalent to the progressive erosion of an active eruption vent (e.g., Fee et al. 2016; McNeil et al. 2018), or the progressive rupture around a repeating subsurface earthquake source (e.g., Park et al. 2019), although the experimental destructive process is not scaled, and little is known about the destructive source processes in natural systems. We regard these aspects of the experiment as the primary contributions to the large observational changes event-to-event. It is clear that care must be taken before ascribing source stability, or progressive variations to one process or another (e.g., Park et al. 2019).
Source directivity
Moving next to the systematic aspects of the frequency content for each event, we first examine if we have completely sampled the spectral content and, if not, what is the upper frequency that can be faithfully interpreted? For several seismic studies in volcanic settings, the interpretation is undertaken in the 2–25 Hz range for data sampled at 100 Hz (e.g., about 50% of the corresponding Nyquist frequency of 50 Hz) (e.g., Hotovec et al. 2013). In volcano observatory settings, helicopter and fixed wing aircraft noise is also commonly observed at frequencies > 10 Hz, and detailed assessments have resolved persistent signals as high as 35 Hz (Eibl et al. 2015) corresponding to about 70% of Nyquist. From this, one could anticipate that frequencies of ~ 70 Hz might be resolved in our 200 Hz data. However, frequency characteristics of a more persistent ‘tremor like’ source (e.g., a helicopter) may be easier to resolve than the short-duration impulses recovered here. Higher frequency observations may correspond to very few data samples for a given waveform in our specific case. In addition, higher frequencies might be impacted by digitizer related anti-alias filters. To assess this, we compared gain-corrected observations to poles-and-zeros restitutions to assess the high-frequency roll-off in the data (Additional file 2: Figs. S2, S3). Inspection of waveforms and spectra shows that amplitudes and frequency distortions are only seen at frequencies < ~ 5 Hz, as expected from the nominal response curve (Additional file 2: Fig. S4).
These considerations suggest that the frequency content of near-field acoustic sensors may be interpreted with reasonable confidence below about 70 Hz. The observed enrichment-depletion occurs around 40–70 Hz for our highest quality observations (Experiments 5, 9, 10, 11 and 12 in Fig. 2, and Figs. 4, 5, 6) while observations above 70 Hz (e.g., Experiment 5-Fig. 5g) are probably on the edge of our ability to interpret. We emphasize here the small number of experimental observations and the high variability observed event-to-event; hence a strong interpretation of the results is not possible in our case. With these caveats, we next consider the range of mechanisms that might produce the frequency enrichment observations.
The overall source dynamics for acoustic sources has been developed from early work by Woulff and McGetchin (1976) and Lighthill (1978) and consists of monopole (explosive) sources, dipole (bi-directional) sources and multipoles (superimposed combinations of dipole sources). Combinations of these source types are generally consistent with volcanic eruptions (e.g., Iezzi et al. 2019; Kim et al. 2012) and are likely to match the observations for our experiments. In this context, we may consider the outcomes of our experiments with equivalent source representations that approximate monopole and dipole source processes (e.g., a complex source mechanism) related to directed explosions. Extending from this, we consider two possibilities for the azimuthal enrichment: (1) frequency enhancement due to a Doppler shift of an extended source process (the source and its extended plume), or (2) a diffraction of acoustic wave energy due to the obstruction of the cannon barrel and trailer bed. Other contributions to variable source spectral observations might include variable water height in the inclined barrel or systematic shifts in the explosive source at the base. Such effects are hard to assess within this contribution and are not considered further.
Plume energetics and the Doppler shift
In the first case, we assume that the acoustic source propagates at subsonic speeds and that the source includes both the initial rupture and expansive discharge, but also a distributed source component representing the ejected plume. The acoustic source is confined and directed from the barrel itself and propagates outward with an efficiency related to the visco-elastic barrel base and walls as well as the opening. We envisage that the barrel is the primary source, but the mass of the ejected fluids may produce acoustic signals by plume turbulence (e.g., Matoza et al. 2009, 2013). From an observational perspective water, N2 gas and colored balls are clearly directed by the barrel orientation. The system may be analogous to water/gas explosions in laboratory settings (Ichihara et al. 2009; Zhang et al. 2016) and also for acid lake hydrothermal eruptions (e.g., Caudron et al. 2018; Jolly et al. 2018). We hypothesize that this system would produce enriched frequencies in the direction of plume propagation, and depleted frequencies behind the plume.
To test this hypothesis, we examined the discrete motions of discharged particles using methods outlined in Jolly et al. (2016). For this purpose, we tracked both individual-colored balls contained in the water as well as individual steam propagation fronts using video data. For the balls, the estimated velocity was about 10–20 m/s while the water splash and N2 gas mixture was found to have a velocity around 28–60 m/s. We note that the balls have lower density and larger volume compared to the individual water splashes mixture, producing increased frictional drag that likely contributed to their observed lower maximum velocities. It is also worth noting that the balls tended to be measured farther along their trajectories away from the source, whereas the water gas jet was measured closer to source (see supplementary videos and Table 3).
Restricting our analysis to the steam propagation and assuming that the plume is the source of mass propagation producing the full infrasound waveform, we use the standard formulation for the Doppler shift, Ft = fo(C + Vr)/(C − Vs), for propagation towards the observation point, and Fa = fo(C − Vr)/(C + Vs), for propagation away from the observation point. Here fo is the frequency of the source, Vs is the velocity of the propagating source, Vr is the velocity of the stationary receiver (0 m/s), C is the acoustic velocity which is assumed 345 m/s. Because the angled source includes both vertical and lateral velocity components, we apply a Cartesian correction to obtain the lateral component along the Earth’s surface. For the maximum amplitude for Experiment 12 (fo ~ 45 Hz at the near source station AC32) we obtain Fa = 43.5 Hz and Ft = 46.6 Hz. The observed range of frequencies is 43 Hz (blue lines in Fig. 3d) and 73 Hz (red lines in Fig. 3d). While the theoretical Doppler shift is of the correct polarity and may partially contribute to the observations, the Doppler-related frequency distortion is not sufficient to produce the acoustic observations in Fig. 3. While it is likely that the discharge velocities within the cannon barrel are more than those observed in the video, a Doppler shift cannot fully explain the observations for Experiment 12.
Wave diffraction and frequency content
An alternative hypothesis is that the cannon and trailer may produce a natural barrier to the propagation of acoustic energy within the sampled frequency range (e.g., Kim et al. 2012). For a vertical discharge, all frequencies are propagated uniformly, producing the observations shown in Fig. 3c. With increased lateral discharge, the sensors open to the barrel would have no barrier to the full spectral content of the discharge, while sensors laterally and behind the cannon would record diffracted acoustic signals, with these effects being more pronounced at higher frequencies. In this case, the laterally directed energy from the barrel would produce some recoil energy that is partially absorbed and distributed by the barrel walls, trailer, and other elements like rubber wheels, shock absorbers and the ground. We observed recoil and barrel bounce associated with all discharges, and distinct late arriving acoustic transients at the near source station (see Additional file 2: Fig. S5). While we were able to document lag times from the video records that may relate to pressure transients at the near source sensor (AC32), a lack of absolute timing for video records implies inexact measurement of lag times. Regardless, we find lag times from 0.2 (experiment 5) to 0.9 s (experiment 9) after the visible initiation of the experiment.
It is difficult to test the wave diffraction hypothesis, without application of synthetic waveform modeling incorporating elastic or visco-elastic boundaries. While modeling for directed sources in active volcanoes has been completed based on a stationary source with directionality represented by a force vector (e.g., Iezzi et al. 2019), such modeling would be difficult to implement in our case given the scale issues and complexity of the apparatus. Regardless, we surmise that both the extended distributed source model, and the visco-elastic barrier model likely contribute to our observations.