In the wake of recent reports of superluminal acoustic group velocities in sonic and ultrasonic regions of the acoustic spectrum, this paper describes the time-domain manifestation of such group velocities through simulations of the linear propagation of ultrasonic wave packets in a suspension of elastic microspheres. Conditions under which arbitrarily large and negative group velocities can be observed as the speed of a peak in the envelope of an acoustic pulse are described. Propagation simulations demonstrate the physical signature of negative group velocities, as well as the causal compliance of the superluminal acoustic pulses examined in this work.

## I. Introduction

In highly dispersive media, the propagation of broadband ultrasonic pulses is marked by the severe reshaping of the pulse envelope. In describing waves undergoing dispersive propagation, up to five different velocities (i.e., phase, group, signal, front, and energy) have been defined, with each characterizing a specific aspect of the evolving pulse.^{1} In specific situations, one or more of these velocities may not be clearly defined or physically meaningful. The most fundamental of these quantities is the phase velocity $cp$. The phase velocity and attenuation coefficient are the primary components of the complex wave number, which when known for all Fourier components of a signal, can be used to predict the evolution of a pulse in a dispersive linear medium. (Complete knowledge of $cp$ also implies full knowledge of the complex wave number through Kramers-Kronig relations.) The group velocity $cg$ can almost always be defined, although its direct physical significance or utility is not always apparent. The group velocity does have some intriguing properties, not the least of which is the fact that it can take on both arbitrarily large and negative values. (According to convention, group velocities such that $cg\u2aa2cp$, or $cg<0$ are collectively labeled “abnormal”.) This paper examines the abnormal regime of the group velocity spectrum in the context of time-localized ultrasonic waves propagating in suspensions of elastic microspheres. The goals of this work are to demonstrate the laboratory signature of negative acoustic group velocity through simulations of wave packet propagation through this two-phase medium, where the solid polymer spheres and the liquid host (saline) have negligible viscous losses and intrinsic attenuation. The transport properties of the microsphere suspension are built upon solutions for acoustic scattering from a sphere in liquid of Faran^{2,3} together with the multiple scattering wave number of Waterman and Truell.^{4} This suspension model has been shown to accurately describe the attenuation, phase velocity, and backscattering properties of agitated polymer microspheres in saline in the low-MHz $(3\u201330MHz)$ ultrasound band^{5,6} at solids volume concentrations around 1%. Recently, negative group velocities have been measured in such a suspension at volume fractions of 3%.^{7} Simulations of ultrasonic pulse propagation in the suspension demonstrate how negative group velocities are physically manifested in the laboratory and the compatibility of such velocities with causality.

Abnormal group velocities are a general feature of wave propagation in dispersive media. The abnormal regime has been most fully explored with electromagnetic waves, chiefly in the microwave and optical regions of the spectrum. The idea that electromagnetic group velocities could exceed the speed of light in vacuum, $c$, was appreciated in the early 20th century.^{1} However, it was not clear until many decades later that superluminal group velocities (i.e., $cg>c$, or $cg<0$) do have a physical manifestation as a pulse velocity^{8} and that such velocities are consistent with relativistic causality. The term superluminal implies that the time interval marking the appearance of some point-of-reference at two distinct locations is less than the time it would take light to go the same distance in vacuum. This definition naturally includes negative velocities since they describe an “object” that goes a positive distance in a travel time that is less than zero. Since the initial observation for propagation in a linear medium,^{9} electromagnetic and electronic pulses with superluminal group velocities have been detected in a variety of systems.^{10–13} It has been previously recognized that superluminal acoustic velocities are possible in dispersive media,^{14,15} and recently measurements of superluminal velocities of acoustic waves have been reported for sound through an air-filled acoustic filter^{16} and ultrasound in an aqueous suspension.^{7} It should be noted that the signal velocity (i.e., speed of information) is never superluminal, and in this work the leading edge of the pulse is shown to move at an ordinary acoustic speed.

## II. Theory

The dispersive propagation of an acoustic wave packet in a passive linear medium can be described by a transfer function of the form, $H(\omega ,\Delta z)=exp[iK(\omega )\Delta z]$, where $K(\omega )=i\alpha (\omega )+\omega \u2215cp(\omega )$ is the complex wave number, $\omega $ $(=2\pi f)$ is the frequency, and $\Delta z$ is depth of the observation plane in the medium (e.g., position of a microphone or transducer). The complex wave number describes the dispersive properties of the system and is defined in terms of the phase velocity $cp(\omega )$ and attenuation coefficient $\alpha (\omega )$.^{17} If $ReK(\omega )$ is expanded in a Taylor series, the first-order expansion coefficient defines the group velocity

For time-localized signals it has been shown that $cg(\omega )$ is the velocity of the envelope of a sufficiently narrowband Gaussian-gated continuous wave.^{8} Numerical work suggests that this physical manifestation of $cg(\omega )$ as the speed of a peak holds for many singly peaked, smoothly enveloped (e.g., Hann windowed) narrowband signals. The key condition is that the bandwidth of the signal must be sufficiently narrow that the wave number can be accurately represented by its second-order Taylor expansion over that frequency range.

Derived from Eq. (1), the conditions for the existence of superluminal and negative group velocities in dispersive media are

The transport model for the microsphere suspensions is built around Faran’s solution^{2,3} for the scattering amplitude of elastic spheres in an inviscid fluid

where $cw$ is the velocity in the fluid. The set of coefficients ${Dn}$ depends on the wave-number-radius products for the longitudinal mode in the sphere material, for the shear mode in the sphere material, and for the acoustic mode of the fluid, as well as the densities of the fluid and sphere material. With knowledge of the distribution of scatterer sizes, the acoustic complex wave number for a suspension of these spheres can be determined. Using Eq. (3) and a distribution of sphere radii, the Waterman-Truell multiple-scattering wave number^{4} for the coherently propagating wave in the suspension takes the form

where $\sigma \epsilon q$ is the volume concentration of spheres of radius $aq$. The quantity $\sigma \u2261\u2211q\sigma q$ is the total volume fraction of spheres and $\epsilon q\u2261\sigma q\u2215\sigma $ is the normalized volume fraction for each sphere radius. [Note that the Waterman-Truell result is formulated in terms of far-field scattering amplitudes but it is derived for scatterers of any separation, that is, there is no restriction on the minimum separation of the scatterers. The appearance of the far-field amplitude in the wave number is a result of the derivation and not an assumption. This result has recently been rederived with another approach.^{18} Experimentally, the Waterman-Truell result has been shown to describe phase velocity accurately at volume fractions beyond those of interest in this work (e.g., see Ref. 19)].

The microsphere size distribution used here was experimentally determined as part of a previous study.^{6} The distribution has a volume-weighted mean radius of $50.5\mu m$ and a full-width at half maximum of $4.8\mu m$. All further references to a suspension refer to one with the same normalized distribution of sphere radii ${\epsilon q}$.

Spectral bands that can support negative delays occur where $dcp(\omega )\u2215d\omega >0$ (known as regions of anomalous dispersion in optics). Figure 1(a) shows the group delay for three volume fractions. The suspension has three bands for which negative delays are possible. The attenuation coefficient and phase velocity of the suspension are shown in Figs. 1(b) and 1(c). The three bands defined by $dcp(\omega )\u2215d\omega >0$ are identified by the pairs of vertical lines in each plot. For the first band, a volume fraction of 10% is sufficient to push the majority of the group delay curve into the negative regime. In Fig. 1(d), the minimum group delay is plotted as a function of volume fraction (occurring between 7.1 and $7.3MHz$).

## III. Results

For a demonstration of the direct observation of negative group delay, consider the simulated immersion tank experiment illustrated in Fig. 2. The suspension is separated from the water by a thin planar barrier that is acoustically transparent. Two transmitting transducers are placed side-by-side in the water, each facing normal to the water-suspension interface. Each transmitter is axially aligned with a receiving transducer in the suspension. The two receiving transducers are placed at different depths in the suspension relative to the boundary. The transmitters generate identical pulses, and the only differences in the received pulses are due to the dispersion over the extra depth of travel to the distal (i.e., more distant) receiver. The signal captured by each receiving transducer is displayed by an oscilloscope. The initial signal from each transmitter is a continuous wave gated by a Gaussian window. The Gaussian window is cut off symmetrically at both ends to isolate the signal in time, and this truncated Gaussian curve is downshifted in amplitude so it goes to zero more smoothly. The propagation is simulated using a Fourier domain approach that utilizes the transfer function with the acoustic complex wave number of (4). The magnitudes of the analytic signal for each waveform are calculated and used for the signal envelopes. The volume fraction of spheres is 0.107 and the frequency used is $7.14MHz$, where the greatest negative delay occurs. The group delay curve for this volume fraction is shown in Fig. 1(a). Figure 3 shows the results of the simulations with the two receiving transducers offset by $0.30mm$ at various depths in the suspension. In Fig. 3(a) the proximal (i.e., shallower) transducer is just inside the suspension. In Figs. 3(b)–3(f) the depth of the proximal transducer ranges from 0.3 to $1.8mm$.

The propagated signals shown in Fig. 3 consist of two competing elements, the primary signal, which propagates with a negative group delay, and the transient response. At zero depth, the signal is the primary Gaussian-enveloped continuous wave described earlier [see the proximal signal in Fig. 3(a) for its shape]. The frequencies at which negative delays occur for the 0.107 volume fraction are also strongly attenuated; thus the amplitude of the primary signal is sharply reduced with depth as most of its Fourier energy falls within a strong attenuation band. The transient signals are generated at the front and back edges of the pulse due to the nonanalyticity at those points. The component frequencies of the transients are largely those outside the attenuation bands and so are much less attenuated with depth than the primary signal. At shallow depths, the primary signal is the dominant feature in the captured waveform, but as the receivers move deeper the transient responses will begin to dominate as they suffer relatively little attenuation. The primary and the transients also move in different directions in time as a function of depth. With increasing depth, the front edge transient effectively sweeps away the remaining primary signal; at that point and beyond, the waveform delays are positive and the negatively propagating groups are no longer observed. In this manner, causality is maintained.

In order to clearly visualize the laboratory signatures of negative group velocities, several movies of propagation simulations (Mm. 1–5) are provided. All of these simulations refer to the experimental setup described earlier and shown in Fig. 2. For the first set of simulations, both of the transducers remain a fixed distance apart and they are both moved progressively deeper into the suspension with each frame of the movie. The wave forms are those captured by the two receiving transducers. The magnitudes of the analytic signal for each wave form are calculated and used for the signal envelopes. For the proximal receiver (shown in green in the simulations) only the signal envelope is shown so that the wave form of the distal receiver can be clearly visualized. In Mm. 1, the case of a “normal” positive valued group velocity (i.e., $cg>0$, $\u223ccp$) is shown in order to provide a point of comparison. As the transducers move deeper into the suspension, the time ordering of the two signals maintains the usual relationship, with the signal from the deeper transducer always occurring later in time. Mm. 2, however, shows the simulation for a negative group velocity. Now the peak of the signal from the distal transducer occurs earlier in time until it is attenuated away, leaving only the transient signals from the pulse edges. These transient signals keep the “normal” time-ordering with those from the proximal transducer occurring earlier.

A variation on this experiment is to keep one transducer fixed in depth just inside the suspension while continually moving the other receiver deeper into the suspension. The results of these “fixed-reference” simulations for the normal group velocity are shown in Mm. 3, and the negative group velocity case is shown in Mm. 4 and Mm. 5.

## IV. Discussion

The compatibility between negative group delays and causality has been widely noted, and is commonly attributed to the “pulse reshaping” effect. The causal consistency can also be verified using Kramers-Kronig relations between group velocity and attenuation^{20,21} which stem from the strict ordering of cause and effect. The causal nature of the processes observed in this work is illustrated by the signals produced in the simulations, as in Fig. 3, Mm. 2, 4, and 5. As a receiver is pulled deeper into the suspension, the front edge transient signal moves later in time (positive delay) while the peak of the initially Gaussian primary signal envelope moves forward in time (negative delay). However, the primary envelope never moves through the leading transient to earlier times; rather it is attenuated beyond recognition before its peak can reach the leading edge of the signal. The leading edge transient, which propagates with a “normal” acoustic velocity, reaches all depths first and so its velocity is effectively the signal velocity. If the primary envelope were somehow able to remain intact and pass through the leading transient to earlier times, then there might be something extraordinary at work. However, the envelope effectively rides on top of an ultrasonic tone burst moving at an “ordinary” acoustic velocity, and if parts of the envelope outrun this support structure, they cease to exist.

In summary, the physical signature of negative acoustic group velocity in a suspension of plastic microspheres has been shown through propagation simulations. The negative velocity of the waveform envelope is countered by the positive velocity of the transients at the edges of the pulse. Beyond a sufficient depth, the transients come to dominate the signal as attenuation effectively extinguishes the original pulse. The negative velocity group never overtakes the leading edge transient and causal propagation is maintained.