Immersive Wave Propagation Experimentation: Physical Implementation and One-Dimensional Acoustic Results

We present a fundamentally new approach to laboratory acoustic and seismic wave experimentation that enables full immersion of a physical wave propagation experiment within a virtual numerical environment. Using a recent theory of immersive boundary conditions that relies on measurements made on an inner closed surface of sensors, the output of numerous closely spaced sources around the physical domain is continuously varied in time and space. This allows waves to seamlessly propagate back and forth between both domains, without being affected by reflections at the boundaries between both domains, which enables us to virtually expand the size of the physical laboratory and operate at much lower frequencies than previously possible (sonic frequencies as low as 1 kHz). While immersive boundary conditions have been rigorously tested numerically, here we present the first proof of concept for their physical implementation with experimental results from a one-dimensional sound wave tube. These experiments demonstrate the performance and capabilities of immersive boundary conditions in canceling boundary reflections and accounting for long-range interactions with a virtual domain outside the physical experiment. Moreover, we introduce a unique high-performance acquisition, computation, and control system that will enable the real-time implementation of immersive boundary conditions in three dimensions. The system is capable of extrapolating wave fields recorded on 800 simultaneous inputs to 800 simultaneous outputs, through arbitrarily complex virtual background media with an extremely low total system latency of $200\mu \mathrm{s}$. The laboratory allows studying a variety of long-standing problems and poorly understood aspects of wave physics and imaging. Moreover, such real-time immersive experimentation opens up exciting possibilities for the future of laboratory acoustic and seismic experiments and for fields such as active acoustic cloaking and holography.

Popular Summary

Seismic waves propagating through Earth’s interior allow scientists to study the plane’s internal structure. Moreover, advances in high-performance computing enable the simulation of seismic wave propagation with ever-increasing physical and structural complexity. However, gaps still exist between field observations and computer simulations. Laboratory seismic experiments can play a crucial role in bridging such gaps, but they suffer from a number of limitations. We present a fundamentally new laboratory that circumvents some of these limitations by linking a physical experiment with a computer simulation in real time, thereby fully immersing the physical experiment in a virtual world.

The smallest seismic wavelengths are considerably larger than a laboratory, contaminating the signal of interest in the form of undesired wave field reflections at the boundary of the laboratory. This is commonly overcome by increasing the frequency of the probing signal, thereby reducing its wavelength and duration, which allows signals of interest to be isolated. However, as the wave propagation in Earth materials is significantly frequency dependent, this can impede the understanding of real-world observations.

Our laboratory features a high-performance computing engine that predicts the wave field at the laboratory boundary, emits a wave field that destructively interferes with the undesired reflections, and simulates the interactions with the virtual surroundings. This suppresses reflections from the laboratory boundary and allows the study of realistic Earth materials at frequencies closer to those observed in the field.

Our setup allows the study of a variety of long-standing problems and poorly understood aspects of wave physics and imaging. Moreover, such real-time immersive experimentation opens up exciting possibilities for the future of laboratory acoustic and seismic experiments and for fields such as active acoustic cloaking and holography.

Figure 1

Schematic of the physical (right) and the full (left) domains introduced in the IBC method with acoustic medium parameters $\rho$ (mass density) and $c$ (wave propagation velocity). The medium parameters need to be identical in the common region between $S^{\mathrm{rec}}$ and $S^{\mathrm{emt}}$ (shaded area); everywhere else they may differ. The solid blue line (right) represents a rigid boundary, while the dashed blue line (left) depicts a transparent boundary at $S^{\mathrm{emt}}$.

Figure 2

Schematic of the numerical implementation of Eq. (4). (a) The surface integral is replaced by a matrix-vector multiplication of the extrapolation Green’s functions with the normal particle velocity and pressure recorded at $S^{\mathrm{rec}}$. Quantities $N_t$, $N_{\mathrm{AI}}$, and $N_{\mathrm{AO}}$ correspond to the number of time steps and the numbers of input and output channels (or sensors and sources), respectively. Note that the extrapolation Green’s functions are scaled by the separation distance of the sources $\mathrm{\Delta }x$ to account for the integration step size. (b) The resulting column vector with dimensions $N_{\mathrm{AO}}{N}_t\times 1$ is reshaped to a matrix with dimensions $N_{\mathrm{AO}}\times N_t$ (black arrow), which is used in the recursive evaluation of the time-convolution integral to yield the normal particle velocity at $S^{\mathrm{emt}}$. At each time step $k$, the leftmost column of the resulting matrix (green box) is applied as the boundary conditions on $S^{\mathrm{emt}}$. The values for $l>k$ (purple box) are stored for later use.

Figure 3

Demonstration of real-time particle velocity estimation and wave field separation in 1D using a Ricker wavelet with a center frequency of 2 kHz as input. The top panel displays two pressure measurements recorded at ${\mathit{x}}_1^{\mathrm{rec}}$ and ${\mathit{x}}_2^{\mathrm{rec}}$. Pressure and (estimated) particle velocity are extrapolated to ${\mathit{x}}^{\mathrm{rec}}$ by evaluating Eq. (4) in real time on the WaveLab system. The scaling by the acoustic impedance of air and convolution with $\widehat{\alpha }(k)$ is included in the extrapolation Green’s functions to obtain a real-time estimate of the right-going pressure $\overrightarrow{p}({\mathit{x}}^{\mathrm{rec}},k)$. The left-going pressure $\overleftarrow{p}({\mathit{x}}^{\mathrm{rec}},k)$ is obtained by subtracting $\overrightarrow{p}({\mathit{x}}^{\mathrm{rec}},k)$ from the total pressure field offline. The successful separation of the wave field is underlined by the low residual energy of 0.37% and 1.09% in the left- and right-going arrivals compared to the total field contained in the left and right shaded box, respectively.

Figure 4

Estimated time delay of the direct wave at five locations along the waveguide (dots). The line of best fit (solid line) intersects with the ordinate at approximately 1.3 ms, which represents $t_{\mathrm{IBC}}$. The standard error of regression is 0.01 ms or 0.75%.

Figure 5

Waveguide used for the demonstration of nonreflecting IBCs, where the virtual medium corresponds to a homogeneous half-space.

Figure 6

Panels 1–5 show pressure recordings at $x_1^{\mathrm{rec}}$ for emitted Ricker wavelets with center frequencies from 1 to 5 kHz. Results with an active IBC source (red dashed trace) are compared to the reference with an inactive IBC source (blue solid trace). The first reflections from the right tube boundary are contained in the grey boxes. Numbers represent the energy (sum of the squared pressure values) in the window relative to the reference trace. Panel 6 depicts the energy in the window relative to the reference as a function of frequency, summed over all five experiments.

Figure 7

Top: Geometry of the three-layer waveguide. Time-distance panel (1): Reference pressure field resulting from the emission of a Ricker wavelet with 2-kHz center frequency with inactive IBCs. Panel (2): As above, but with an active IBC source on the right. Panel (3): Difference of both pressure fields. Positions immediately next to the diameter changes and recording locations do not contain data due to practical limitations.

Figure 8

Top: Geometry of the homogeneous waveguide. A perfectly rigid virtual reflector 30 cm away from the physical termination of the waveguide (orange dashed line) is included in the extrapolation Green’s functions. Time-distance panel (1): Reference pressure field obtained by emitting a Ricker wavelet with 2-kHz center frequency with inactive IBCs. Panel (2): Wave field with an active IBC source. Positions immediately next to the recording locations do not contain data due to practical limitations.

Figure 9

High-level view of the WaveLab system architecture. From top to bottom: The input data of 800 channels are collected over four analog input (AI) chassis and aggregated on the AI data master. The full input data set is distributed to six analog output (AO) data masters, which further pass the data set to 29 analog output chassis. Each AO chassis consists of up to 14 compute nodes, each capable of computing the output for two sources. The start of an experiment is triggered and synchronized by a central timing and stimulus chassis (blue dashed lines).

Figure 10

Photograph of the full-scale acquisition, computation, and control system.

Figure 11

Setup of the 1D sound wave tube experiment: (1) and (2) stimulus and IBC sources (B&C DE-7 loudspeaker); (3) two-channel source amplifier (Monacor SA-100); (4) interrogation microphone and preamplifier (PCB Piezotronics 377C13 and 426E01); (5) IBC microphones (PCB Piezotronics 377B13 and Brüel & Kjær 4947) and preamplifiers (Brüel & Kjær 2671) with a separation of $d=1.8\mathrm{cm}$; (6) four-channel signal conditioner and microphone amplifier (PCB Piezotronics 482C15); (7) and (8) 3D printed, air-filled waveguides of 2- and 4-cm diameter, respectively.

Figure 12

Demonstration of the manipulation of the Green’s functions for the extrapolation from two pressure recordings at $x_1^{\mathrm{rec}}$ and $x_2^{\mathrm{rec}}$ to one source at $x^{\mathrm{emt}}$. During the extrapolation, the extrapolated values for the recordings on channels ${\mathrm{AI}}_1$ and ${\mathrm{AI}}_2$ are added, which concludes the particle velocity estimation and pressure interpolation. The chosen extrapolation Green’s functions only act as an example and are not related to the experiments presented in this study.

Figure 13

Device path from analog input to analog output (panel 1) and Low-level view of the WaveLab system architecture (panel 2). Green and blue arrows represent data transfer and trigger signals, respectively. The numbers in parentheses and in the device path correspond to the National Instruments module names. The hardware components and device path used for the 1D experiments demonstrated in this study are denoted by red dashed lines.