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Dynamic response and hydrodynamics of polarized crowds

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Science  04 Jan 2019:
Vol. 363, Issue 6422, pp. 46-49
DOI: 10.1126/science.aat9891
  • Fig. 1 Hybrid-wave propagation in queuing crowds.

    (A) Picture of the starting corrals of the Bank of America Chicago Marathon (2017) taken from an elevated observation point (32). Thousands of runners progress toward the starting line under the guidance of race staff members. (B) The chain formed by the race staff advances with repeating sequences of walks and stops. (C) Velocity and density fields at three successive times. At Embedded Image, the crowd is static and has a uniform density Embedded Image (blue lines). At Embedded Image, as the staff members displace the downstream boundary of the queuing crowd, a hybrid wave packet that couples velocity and density fluctuations propagates upstream. Embedded Image indicates the distance from the starting line located at Embedded Image.

  • Fig. 2 Underdamped propagation of linear and nondispersive velocity waves.

    (A) Kymograph of the longitudinal velocity, averaged over the transverse direction (Chicago 2016). Embedded Image indicates the distance from the starting line. A number of velocity waves are seen to propagate upstream at the same speed. (B) Probability distribution function of wave speed, measured for all the studied events. The typical wave speed hardly differs from one event to the other. The overall speed distribution is narrowly peaked around Embedded Image. A., Atlanta; C., Chicago; P., Paris. (C) Black line: velocity of a chain of race-staff members Embedded Image, measured by direct tracking. The corresponding positions Embedded Image are reported as a black line on the kymograph in (A). As illustrated with the same color code on the kymograph (A), the colored curves correspond to the longitudinal velocity field measured along the curves defined by the race staff position Embedded Image after four different waiting times Embedded Image : Embedded Image. Independently of the shape of the Embedded Image signal, the velocity waves faithfully propagate the information of the boundary speed Embedded Image over system-spanning scales, at constant speed.

  • Fig. 3 Spectral properties of speed waves in queuing crowds.

    (A) Probability distribution function of the longitudinal and transverse components of the velocity field (Chicago 2016). The longitudinal component dominates. (B) Power spectrum of the flow speed, plotted for Embedded Image. The spectrum is normalized at every wave vector Embedded Image. The inset represents a normalized speed spectrum for a crowd in free-flow conditions. Data are from the Chicago 2016 experiment. (C) Variations of the celerity of the speed waves with the direction of the wave vector θ for all experiments. Circles represent experimental data. The solid line represents cosine fit. The excellent fit shows that the dispersion relation is given by Embedded Image. (D) Normalized speed correlations plotted versus time for all experiments (Embedded Image, wave vector Embedded Image). (E) Damping rate of the speed waves, α, plotted for all wave vectors along Embedded Image. Circles represent experimental data. Solid lines represent best quadratic fits. (F) Variations of the speed diffusivity Embedded Image with the direction of propagation. Circles represent experimental data. Solid lines represent squared cosine fits. (G) Normalized orientational correlations plotted versus time for all experiments (Embedded Image, Embedded Image). (H) Damping rate of the orientational fluctuations Embedded Image plotted as a function of the wave vector (Embedded Image). Circles represent experimental data. Solid lines represent best quadratic fits Embedded Image, with Embedded Image. (I) Inverse relaxation time Embedded Image plotted as a function of the direction of the wave vector θ. No significant angular variation is observed.

Supplementary Materials

  • Dynamic response and hydrodynamics of polarized crowds

    Nicolas Bain and Denis Bartolo

    Materials/Methods, Supplementary Text, Tables, Figures, and/or References

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    • Materials and Methods
    • Supplementary Text 
    • Captions for Movies S1 to S3
    • References

    Images, Video, and Other Media

    Movie S1
    Start of the 2016 Chicago Marathon, as seen from an elevated observation point. Thousands of participants progress towards the starting line in a 200 m long and 20 m wide start corral.
    Movie S2
    Same as Supplementary Movie S1, after we corrected for perspective distortion and measured the velocity field (color codes for longitudinal velocity). Velocity waves propagate upstream over the whole extent of the corrals.
    Movie S3
    Two-seconds animated GIF of a part of the static crowd (Chicago 2016). White circles indicate detected participants on a static image. Yellow circles show some participants who were not visible on the static image.

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