Evidence of wave front folding of sonic booms by a laboratory-scale deterministic experiment of shock waves in a heterogeneous medium
(Color online) The experimental setup. A , plane wave is emitted by an array of 256 transducers (right) so as to create a well-formed sawtooth shock wave at twice the shock formation distance . Plane wave front is achieved, thanks to the inverse filter technique and programmable electronic control (128 independent channels). The plane shock wave impinges a heterogeneity (center) made of a semicylinder of silicone playing the role of an acoustical lens with a diameter comparable to the acoustical wavelength and supported by a silicone plate. The scattered pressure field is scanned behind the heterogeneity by a movable, wide-band, calibrated hydrophone.
Measurement of the coefficient of attenuation with the membrane hydrophone (dashed line), with the optical interferometer (dotted line) and the theoretical fit (solid line).
Calibration of the membrane hydrophone: incident pressure measured with the uncalibrated membrane hydrophone (dashed line), with the constructor’s calibration (dotted line) and with the optical interferometer (solid line).
The spatiotemporal representation of the incident shock wave (top) and the temporal signal of the incident shock wave (bottom) measured just before the heterogeneity.
The spatiotemporal representation of the pressure field measured at the focus of a heterogeneity (up) and pressure measured at the center of the focusing (bottom).
Time dependence of the pressure measured before the heterogeneity (solid line), behind a plane silicon plate (dotted line) and at the focus (dashed line).
The spatiotemporal representation of the pressure field (a) before a heterogeneity, (b) at the focus, (c) at , and (d) at . [(e)–(h)] Time dependence of the pressure on the axis at the same distances. [(i)–(l)] Time dependence of the pressure off axis at the same distances.
Spatiotemporal representation of the pressure field measured behind (a) and (c) heterogeneities. Corresponding time dependence of pressure off axis behind (b) and (d) heterogeneities.
Time dependence of pressure measured before (dash-dotted line) and behind a (solid line), (dashed line), or (dotted line) heterogeneity.
Different waveforms measured with a single heterogeneity of different sizes (—first column, —second column, and —third column). The measurement points are located on the axis and at the focus of the largest heterogeneity (first row), beyond the focus and off axis (second row) and beyond the focus and off axis (third row).
Example of sonic boom time waveforms from the BoomFile database (Lee and Downing, 1991).
The spatiotemporal representation of the pressure field measured (up) and simulated (bottom) at the focus of a heterogeneity.
Time dependence of pressure measured (dashed line) and simulated (solid line) on the axis of propagation at the focus behind a heterogeneity.
Pressure measured (up) and simulated (bottom) behind a heterogeneity at for several off axis distances (solid line: , dashed line: , and dotted line: ).
Difference between the pressure frequency spectrum, measured at the focus of a heterogeneity (normalized by the incident pressure) in nonlinear and linear propagations, from .
(Color online) Simulated amplitude of the harmonics of the pressure frequency spectrum at the focus (axial point of maximum peak overpressure) vs the diameter of the heterogeneity. Comparisons between linear (solid lines) and nonlinear (dashed lines) numerical simulations. Harmonic amplitudes are normalized by the amplitude of the fundamental of the incident pressure field. Heterogeneity diameter is normalized by the fundamental wavelength . From left to right and top to bottom: first to eighth harmonics.
Numerical simulation of the axial waveform rise time vs the axial distance . Rise time is defined as the time interval separating the two times at which pressure time waveform at a given point is equal to 90% of, respectively, the minimum and the maximum pressure values. Rise time is normalized by the rise time of the incident shock wave (equal to ). Distance is normalized by the diameter of the heterogeneity. Simulations are shown for a heterogeneity with diameter (solid line), (dashed line), (dash-dotted line), or (dotted line).
Comparison of different laboratory-scale sonic boom experiments.
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