Characteristics of a typical ‘‘ N -wave’’ sonic boom measured on the ground. ͑ a ͒ Simple N -wave time series. The straight-line approximation is parameterized by a ‘‘rise time’’ ͑ t ͒ which is the time from the onset of the boom to maximum pressure, and the total duration ͑ D ͒ of the waveform. Rise times typically range from 2–20 ms, and durations are typically 100– 400 ms. ͑ b ͒ Theoretical energy spectrum of an N -wave boom with a rise time of 8 ms and a duration of 350 ms. 

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... traveling faster than the speed of sound generate shock waves that result in impulsive acoustic signatures known as sonic booms. The typical acoustic signature of a sonic boom is the ‘‘ N -wave’’ ͓ Fig. 1 ͑ a ͔͒ , which is characterized by sharp pressure jumps at the front and back of the waveform, with a slow pressure drop in between. It has been recognized from the early days of supersonic flight that sonic booms generate undesirable environmental impacts over populated areas, 1 primarily because of startle response to the shock wave pressure rise, and low-frequency building response ͑ i.e., vibration, rattle ͒ . The undesirable acoustic qualities of sonic booms led to legislation in the U.S. ͑ and most countries internationally ͒ forbidding supersonic flight and the generation of sonic booms over land, except in designated military corridors. As a result, most sonic booms are currently generated over the ocean. Sources of sonic booms over water include the Concorde, which flies routinely between Paris and New York, and rocket launches associated with satellite deployments. The restriction of supersonic flight to air spaces over water has refocused sonic boom environmental impact research to the marine habitat, and to marine mammals, in particular. While the characteristics of sonic booms in air are well understood and supported by a vast body of research ͑ e.g., Carlson and Maglieri, 2 Darden 3 ͒ , data constraining the penetration of sonic booms into the ocean, and the characteristics of boom pressure signatures underwater, are scarce. The original theory for the propagation of sonic booms across the air/sea interface was developed by Sawyers, 4 and by Cook. 5 For level flight, booms generated by objects traveling at speeds less than that of sound in water ͑ 1500 m/s, or Mach ϳ 4.4 ͒ create an evanescent wavefield in the water column, decaying exponentially with depth. The decay is wavelength dependent, with short wavelengths ͑ high frequencies ͒ attenuating faster than long wavelengths ͑ low frequencies ͒ . The Sawyers and Cook theories were validated with laboratory experiments involving spherical blasts 6 and small, high-speed projectiles. 7 Early attempts to validate the theory with field experiments, however, were unsuccessful. Young 8 and Urick 9 attempted to quantify the penetration of sonic booms in the ocean ͑ in separate experiments ͒ by measuring boom pressure signatures immediately above, and at several depths below, the air/sea interface. Underwater sonic boom pressure measurements from these experiments exhibit different decay rates with respect to depth, and neither matches the analytical theory or laboratory data. Urick’s results devi- ated enough to cause him to question the validity of the evanescent wave theory for sonic booms in water. Disagreement between the field data and the analytical theory introduced some uncertainty regarding the validity of the theory and its underlying assumptions in real world as opposed to laboratory conditions. In particular, the theories of Sawyers 4 and Cook 5 both assume a perfectly flat ocean surface, but the ocean surface is continually perturbed by ocean waves. The possible effects of a realistic ocean roughness on the penetration of booms into the water was recently investigated using numerical methods by Rochat and Sparrow, 10 and Cheng and Lee, 11 with each arriving at different conclusions. Rochat and Sparrow concluded that roughness has a negligible effect, with underwater pressure level variations from a flat interface of 1 decibel or less. In contrast, Cheng and Lee concluded that the sea surface roughness exerts a first-order effect on boom penetration, particularly at large depths and low frequencies. At issue is the magnitude of the scattered component of sonic boom energy in the water column, and its proportion to the evanescent signal. The lack of consistency between the numerical studies, 10,11 and disagreement between the analytic theory 4,5 with the field experiments, 8,9 underscores the need for reli- able measurements of sonic booms underwater to serve as a benchmark for the validation of theoretical models, and to provide a foundation for environmental impact assessments. The early data of Young and Urick suffer from the techno- logic limitations of their day. Specifically, the data acquisition systems employed in the experiments did not have adequate low-frequency response, and the pressure measurements are likely contaminated by the interaction of the sonic boom with the mechanical systems used to suspend the hydrophones in the water column. A sonic boom measured by Urick 9 is shown in Fig. 2. The pressure signatures bear little resemblance to an N -wave, primarily because the measurement system lacked the low-frequency response to measure the slow pressure decay between the fore and aft shocks. In addition, the ringing observed in the water column measurement indicates that the data are contaminated by mechanical interactions with the suspension system. Instrumentation has improved dramatically since the experiments of Young and Urick, and modern systems are ca- pable of making high-fidelity measurements of sonic ...

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... but provided redundancy should the in-air sensor on the data buoy have failed. The SABER measurements ͑ with a sampling rate of 10 kHz ͒ were also used to examine the high-frequency characteristics of the booms en- tering the water column. Six supersonic passes were made with U.S. Navy F-4 aircraft over the two days of the experiment. The overflight altitude was varied from 610–6100 m ͑ 2000–20 000 ft ͒ to provide a range of boom pressures ͑ 96–530 Pa, or 2–11 psf ͒ at the air/sea interface. Aircraft speeds ranged from Mach 1.07–1.26, corresponding to the aircraft’s best speed at each altitude. After transiting from Pt. Mugu to the experimental site, the aircraft established radio contact with the R/V N EW H ORIZON and were given an updated target position. Once a supersonic run was underway, the test pilots noted the speed, altitude, and heading of the aircraft, along with a single position ͑ latitude/longitude ͒ at the beginning of the run. The aircraft position and magnetic heading ͑ accurate to within 3 deg ͒ at the beginning of the run were used to estimate a flight track for each supersonic pass. Since the aircraft did not have GPS data loggers, it is impossible to know the exact lateral distance between the flight track and the data buoy. Based on straight-line flight track estimates, all of the tracks pass over the buoy within the tolerance of the estimates, except for one ͑ Pass 2 ͒ . The straight-line estimate for Pass 2 runs ϳ 1 km west of the buoy. This is significant given that the horizontal error is only slightly less than the aircraft altitude ͑ 1.5 km ͒ . However, no aircraft had an automatic heading-hold system, and therefore the actual flight tracks for all the supersonic runs may have varied from the straight-line estimates. Time series and spectral plots of each sonic boom measured during the experiment ͑ Pass 1–6 ͒ are shown in Figs. 4–9. Data from six hydrophones were recorded during each run. Measurements made during the first day of the experiment ͑ Pass 3, 5, 6; Figs. 6, 8, 9 ͒ recorded hydrophone data in-air, and at 7, 22, 37, 82, and 112 m beneath the sea surface. Measurements made during the second day of the experiment ͑ Pass 1, 2, 4; Figs. 4, 5, 7 ͒ recorded data in-air, and at 7, 22, 37, 52, and 67 m beneath the sea surface. The shift toward shallower depths on the second day was made after it was realized that the booms were failing to generate detectable signals near the bottom of the hydrophone array. The time series data for the in-water measurements are compared with the linear, analytical theory of Sparrow and Ferguson, 13 which is based on the work of Cook. 5 The method assumes a flat air/sea interface, and no interaction with the seafloor ͑ deep water ͒ , but allows for arbitrary boom wave shapes. To generate the theoretical pressure signatures in the water column, the spectrum of the in-air pressure signature was calculated, weighted by an exponential decay with respect to wavelength and depth, and then transformed back into the time domain. A Blackman window was applied to the in-air data segment before calculating the fast Fourier transform ͑ FFT ͒ to reduce Gibbs phenomenon associated with truncating an infinite series. The theoretical waveforms shown in Figs. 4–9 were bandpass filtered ͑ 3–200 Hz ͒ to mimic the analog circuitry of the in-water hydrophones. This filtering slightly distorts the theoretical wave shapes, but is required to provide an equal comparison to the measurements. The amplitude spectrum of the boom waveform and the ambient noise field at each hydrophone channel are shown on the right side of Figs. 4–9. The amplitude spectrum of the ambient noise field represents the FFT of a randomly se- lected segment of data immediately preceding the boom ar- rival. A Gaussian window was applied to both the boom pressure signature and the ambient noise segment prior to estimation of the amplitude spectrum. The primary experimental objective of this work was to make high-fidelity measurements of sonic booms at the air/ sea interface and at several depths in the water column. We begin the discussion by assessing the extent to which this objective was met. We then compare our measurements to theoretical predictions, and discuss the implications of the similarities/differences for the validation of the theory under real ocean conditions. We conclude with a brief review of some remaining issues and unanswered questions regarding the penetration of sonic booms into the ocean. We begin by examining the fidelity of the in-air measurement, which is important considering that it is used as a source function for the theoretical predictions of boom pressures underwater. All of the in-air pressure signatures in this experiment are characterized by fairly simple N -waves, and the amplitude spectra of the in-air signals have the expected shape, with two separate corner frequencies corresponding to the boom duration and rise time compare top right panels of Figs. 4–9 with Fig. 1 ͑ b ͔͒ . The low-pass filter applied to the in-air pressure data ͑ 60 Hz corner frequency ͒ removes any contributions from reflected phases at the microphone. It appears that our in-air measurements adequately characterize the sonic boom impinging the air/sea interface above the vertical hydrophone array, especially at low frequencies, which are of primary importance to this study. Several of the supersonic passes made during the experiment were at fairly low altitudes, and under these conditions individual shocks from the various aerodynamic features ͑ e.g., nose, wings, cockpit ͒ would not be expected to have coalesced into single bow and tail shocks ͑ e.g., Hayes 15 ͒ . Uncoalesced shocks create extra spikes in the acoustic signature. These extra spikes generate relatively high-frequency pressure perturbations that are removed by the low-pass filters, and in these cases we expect that the simple N -waves rendered by the in-air sensor do not perfectly represent the actual booms at the sea surface. Indeed, the SABER measurements made aboard the R/V N EW H ORIZON for the low- altitude passes contain spikes embedded in the N -wave signature ͑ although these measurements also contain spikes from reflections off the ship’s superstructure ͒ . It will be seen in the following section that the failure to capture high- frequency spikes in the in-air measurement is not a significant shortcoming since these features are almost immediately removed from the evanescent wavefield beneath the sea surface. The principle concern for the underwater boom measurements is to keep noise levels on the individual sensors as close to ambient as possible. Achieving low noise levels on a hydrophone array suspended at shallow depths is difficult because the hydrophones are mechanically linked to the mo- tion of the surface wavefield. The suspension system decouples the hydrophones from the jerking of the array by motions of the buoy and from strum ͑ e.g., Sotirin and Hildebrand 16 ͒ induced by current flowing past the array. Inspection of the ambient noise pressure spectra in Figs. 4–9 indicates that our attempts to minimize noise levels on the hydrophone array using vibration isolators and shock cord ͑ Fig. 3 ͒ were fairly successful. The pressure variance of the ambient field on individual hydrophone elements was typically less than 100 Pa 2 in the relevant band from 3–200 Hz. This corresponds to a dynamic head of less than 1 mm of water, root-mean-square ͑ rms ͒ . Ambient noise pressure variance on the deepest phones is especially small, with typical values of 2–5 Pa 2 . In practical terms this resulted in excellent signal-to-noise levels for boom measurements down to about 40–50-m depth. At this depth the amplitude of the boom pressure signal is equal to or less than the ambient levels on the hydrophone array. Additionally, the underwater boom signatures do not contain any ringing as do those measured by Urick 9 or Desharnais and Chapman. 12 Nor does the acoustic field contain any measurable perturbation from the surface buoy. This demonstrates that by using a small diameter spar buoy as a surface mooring for the data acquisition system we avoided contaminating the boom waveform with mechanical cou- pling down the suspension line. By conducting the experiment in deep water, we also appear to have avoided interaction with the seabed. We conclude that the pressure measurements made during the course of this experiment provide accurate renderings of the sonic boom wavefields generated at the instrument array, especially at low frequencies. The agreement between our data and the analytical method of Sparrow and Ferguson, 13 which is based on the theory of Cook, 5 can be observed by comparing the solid and dashed lines in the time series plots of Figs. 4–9. Data and theory are in agreement at all depths and for all booms within the limitations of the signal-to-noise ratio. The signal is above the noise to depths of 37 m for all booms, and to greater depths for lower altitude flights with stronger booms. Examination of spectral attenuation provides additional insight into the agreement between our data and linear theory. The evanescent decay of a sonic boom underwater 4,5 scales as e Ϫ k 0 ␮ z , where k 0 is the wavelength in air divided by Mach number, z is depth, and ␮ ϭ ͱ 1 Ϫ M 2 / W 2 , where M is Mach number and W is the ratio of sound speed in air to water. The correlation between this theoretical expression and our Pass 1 measurements ͑ strongest boom ͒ is shown in Fig. 10. The data follow the theoretical decay curves until they approach the noise floor, at which point ambient noise overwhelms the signal. The agreement between the predicted waveform and the signal measured at the deeper hydrophones precludes the existence of a scattered component of the sonic boom signal propagating as an acoustic wave in the water with an amplitude greater than 4 Pa peak to peak. The largest sonic boom measured in air had a ...