The impact of shallow waveguides on the spectral characteristics of modulated spectral lines in underwater radiated noise produced by ships

The modulation spectrum line spectrum of undersea radiated noise from ships is a significant characteristic in the identification of ship targets. This research presents an analysis of the impact of waveguide dispersion effect and demodulation bandwidth on the intensity of modulation line spectrum, based on the theoretical framework of square demodulation method. The physical mechanism underlying this influence is derived and discussed. The examination of collected data and numerical simulations demonstrates that the extracted modulated line spectrum intensity in an unobstructed environment effectively utilizes the carrier’s energy and generally does not exhibit any abnormal loss during propagation. However, in a shallow sea dispersive waveguide, as the propagation distance increases, a transimission loss anomaly of over 3dB occurs in the modulated line spectrum. The theoretical derivation yields implications that can offer theoretical backing for the reduction of vibration and noise in underwater targets, extraction of faint signals from high-order line spectra in the distant field, and identification of targets.


Introduction
The detection of hydroacoustic signals is a significant aspect within the realm of hydroacoustic technology and a focal point of ongoing study.The effectiveness of passive sonar systems in detecting underwater targets is directly influenced by the performance of signal detection.The source of interest for passive sonar is the noise emitted by ships, known as ship radiated noise.The spectrum of ship radiated noise encompasses a wide range of target parameter parameters and motion information, serving as the primary foundation for detecting undersea targets.One of the parameters that holds significant importance for sonar operators is the "rhythm" of ship noise [1,2].This refers to the modulation spectrum of the underwater radiated noise signal produced by the ship.By analyzing this modulation spectrum, valuable information such as the ship's propeller speed, number of blades on the ship can be obtained.These parameters are crucial for determining the physical characteristics of a stable ship [3,4].Nevertheless, the data on the ship's radiated noise observed through sonar in the ocean waveguide encompasses not only details about different noise sources and the ship's operational conditions and physical characteristics, but is also influenced by the ocean waveguide factors [5][6][7][8].These factors significantly impact the practical utilization of the data for detection and identification purposes.
Researchers have conducted extensive research on the theoretical model and generation mechanism of target radiated noise modulation spectrum [9][10][11][12][13].This includes the development of (quasi-)periodic smooth stochastic models and the automatic detection and extraction method of the modulation line spectrum, which has garnered significant attention [14][15][16][17][18][19].The impact of modulating dispersive waveguides [20][21] on the modulation spectrum of radiated noise has received limited attention in the literature.The fundamental process remains unresolved, and there is a need to enhance the foundational theory in this area.
This paper aims to elucidate the mechanism behind the anomalous attenuation of line spectrum energy in dispersive waveguide amplitude modulation signals through theoretical derivation.The findings of this study can offer a theoretical foundation and justification for enhancing higher-order spectral line spectrum extraction and long-range target recognition technology.

Theoretical modeling and characterisation of the modulation spectrum propagation of shipboard radiated noise.
The mathematical formulation representing the amplitude modulation signal of the ship's radiated noise can be represented as: s represents a ship radiated noise amplitude modulation signal; ) (t n is ambient noise, suppose it is white Gaussian noise that is not related to ) (t s . The act of squaring the wave of the detector as represented by Equation (1) The variable i b represents the modulation coefficient of the newly written modulation function.
The average power of the modulated line spectrum is subsequently determined by IOP Publishing doi:10.1088/1742-6596/2718/1/0120923 L P is the line spectral strength of modulation.To investigate the impact of the waveguide on the transmission properties of the modulated signal, the amplitude-modulated signal source model is initially simplified to a single-frequency modulated carrier with a single frequency at a specific moment in time.The mathematical formula for this simplified model is as follows: in terms of the difference in intensity between the carrier and the modulating signal frequencies.Typically, the value of m a is less than 1.Consequently, the intensity of the differential frequency is lower than that of the carrier frequency by more than 6dB.In a free field, where the acoustic signal propagates according to the spherical expansion, Eq. ( 5) is rewritten as The variable r represents the horizontal distance between the source and the receiving point, while c denotes the speed of sound.By using the process of squaring Equation ( 6) and subsequently subjecting it to a low-pass filtering operation, we are able to get the absolute amplitude of the amplitude modulation line spectrum at r .
The initial term represents the modulated line spectral fundamental frequency, while the subsequent term denotes the line spectral doubling frequency.Additionally, the power requirement indicates that the intensity will diminish exponentially with the fourth power of distance, which contrasts significantly with the propagation characteristics of a single-frequency signal.
In the context of frequency dispersion, it is assumed that the variation in amplitude magnitude of different frequencies at a distance r from the sound source is approximately equal within a narrower bandwidth, denoted as ) (r A disregarding the effect of depth for now, the sound field of the radiated noise in the dispersive waveguide can be determined using Equation ( 5 In the aforementioned equation, the variable ) , ( r f  represents the magnitude of phase fluctuation experienced by the frequency f at a certain distance r .The square detector and low-pass filtering of Eq. ( 8) yields the amplitude-modulated low-frequency line spectrum as The above derivation is the transmission characteristics of the amplitude-modulated signal line spectrum of a single-frequency carrier or smaller bandwidth carrier in a dispersive waveguide, and based on the above equation, the bandwidth increase can be thought of as the superposition of many smaller bandwidths, and the The adjusted line spectrum fundamental frequencies are represented by the first two terms on the right-hand side of Equation (10), while the third term corresponds to the 2-octave adjusted line spectrum created by squaring the detector.The complexity of the frequency-scattered field adjustment line spectrum sound field is greater when compared to the free-field Equation (6).
When waveguides lead When the phases of the first two items in the line spectra of the same subband or separate subband modulation are different, the power spectral intensity of the modulation spectral line is calculated using Equation (10): (13) The initial component on the right-hand side represents the power intensity of the fundamental frequency line spectrum, while the subsequent component corresponds to the intensity of the line spectrum at twice the frequency, which is created by the squared waveforms detected by the sensor.In this context, the variable When equations ( 11) through (12) are satisfied, equation ( 13 When j denotes each frequency in the bandwidth, then ( 13) and ( 14) can be simplified as respectively  is the power of the carrier.Equation (15b) can be considered as an equivalent form of Equation ( 4) when the presence of interline spectral interference is disregarded.By examining the initial modulated line-spectrum fundamental-frequency intensity in the equation (15), it is evident that the disparity in power intensities of the line-spectrum can be a multiple of the bandwidth B in the most extreme scenario.In (c) Figure1.Analysis of modulation line transmission characteristics of ship 1 (a) 1k-2kHz (b) 2k-3kHz (c) 20-60s(8Hz) Figure 2 illustrates the outcomes of the modulated spectral line spectrum transmission analysis, employing the identical methodology as depicted in Figure 1.The subject of investigation in this case is a typical cargo vessel, possessing dimensions of 180 m in length, 28 m in width, and 10.5 m in draught.The vessel was observed to be travelling at a velocity of 14.3 knots, while maintaining a minimum proximity of 115 m from the hydrophone.In the time range ranging from 1 to 50 seconds, there is a distinct and prominent line spectrum observed at around 28 Hz.This observation is further supported by the anomaly depicted in Figure 2(c), which highlights the transimission loss associated with the frequency of the line spectrum during this specific time interval.
Based on the observation in Figure 2(c), it is evident that the line spectral transimission loss for the grocery ship exhibits unusual fluctuations in relation to the bandpass filtering ranges of 1k-2kHz and 2k-3kHz.These anomalous variations display similarities for propagation distances below 0.3km, characterised by values lower than 3dB.As the distance of propagation increases, there is an observed increase and subsequent decrease in the line spectral transimission loss anomaly associated with the bandpass range of 1k-2kHz, reaching a maximum anomaly of 4.8dB.Similarly, the line spectral transimission loss anomaly corresponding to the bandpass range of 2k-3kHz exhibits a decrease followed by an increase, with a maximum anomaly of 6dB. Figure 3 presents the data analysis of a randomly selected sailing vessel obtained from the South China Sea.The vessel was retrieved at a sea depth of 4000m, and the measurement was taken at a reception depth of around 180m. Figure 3(a) and Figure 3(b) depict pseudo-color representations of the DEMON spectra, wherein the spectra were filtered within the frequency regions of 1k-2kHz and 2k-3kHz, respectively.The horizontal axis represents time, while the vertical axis represents frequency.Notably, a prominent line spectrum is observed in close proximity to 5Hz. Figure 3(c) depicts a comparison of the transimission loss anomalies resulting from various bandpass calculations.These anomalies exceed 4 dB for the time period of 1-800s, while the bandpass filtering calculations within the 2k-3kHz range exhibit significantly bigger anomalies, reaching a maximum of 18 dB.To conduct a more comprehensive investigation into the impact of the waveguide on the transmission of the modulated line spectrum, we simulated the transmission loss anomalies at various distances of 8 Hz in diverse environments: free-field, sea-depth of 30 m, and 100 m iso-sonic waveguide.The corresponding results are presented in Figures 4 to 6.The modelling technique does not account for ambient noise, and it assumes that the energy across multiple bandpasses is approximately uniform.
The simulation results depicted in Figure 4 illustrate the anomaly in free-field transmission loss.The red and blue dashed lines correspond to simulations conducted with bandpasses of 1k-2kHz and 2k-3kHz, respectively.It is observed that the transmission loss anomaly tends to approach zero.Figures 5 and 6 depict the anomalies in transmission loss observed within a specific distance range of the shallow sea waveguide.These anomalies were obtained through the utilisation of the Kraken programme.In Figure 5, a comparison is presented, showcasing the effects of bandpass filtering with varying centre frequencies but the same bandwidth.On the other hand, Figure 6 illustrates a comparison of simulation results with different bandwidths.The findings from the simulation analysis of the waveguide, which has a sea depth of 30m, indicate that there is a general inclination for the transmission loss anomalies to escalate as the distance increases.Furthermore, the variation in bandpasses has a negligible impact on the observed outcomes.The transmission loss anomalies for the 100m sea depth waveguide, when subjected to a bandpass filter in the 2k-3kHz range, exhibit notable variations in both positive and negative directions (see Figure 6a).Furthermore, expanding the bandwidth results in a marginal increase in the magnitude of these transmission loss anomalies (see Figure 6b).(a) same bandwidth (b) different bandwidths Based on the aforementioned analysis of measured data and numerical simulation results, it is evident that significant transimission loss anomalies occur in the modulated line spectrum of the target radiated noise within the shallow sea waveguide.Specifically, there is an insufficient utilization of the broadband bandpass energy, resulting in inefficient utilization of the radiated noise power contained in the bandpass.Consequently, the relative intensity of the modulated line spectrum and the continuum spectrum diminishes, thereby reducing the noise immunity of the line spectrum.This phenomenon adversely affects long-distance scenarios.The process of extracting the desired modulated line spectrum is highly disadvantageous.The aforementioned work primarily examines the fundamental principles governing waveguide influence, while refraining from investigating the impact of various waveguides and changes in transceiver depth at this time.

Conclusion
This research presents an analysis of the impact of waveguide dispersion effect and demodulation bandwidth on the intensity of modulation line spectrum, based on the theoretical framework of square demodulation method.The spectrum intensity of the modulated lines retrieved in the free field is demonstrated to fully use the energy of the carrier using theoretical derivations, data processing, and numerical simulations.Moreover, it is seen that these modulated lines are practically devoid of any transimission loss abnormalities.In the context of the shallow sea dispersive waveguide, significant loss anomalies are observed in the propagation of the modulated line spectrum.This leads to a low efficiency in utilising the broadband carrier energy and reduces the noise immunity of the line spectrum, posing challenges in extracting line spectrum over long distances.Moreover, these conditions hinder the detection and identification of long-distance targets.
Given that the aforementioned test data originates from a shallow marine environment, the primary focus of the modelling efforts is directed towards the shallow sea.The forthcoming study aims to investigate the characteristics of line spectra in deep-sea transmission under typical conditions and propose a method to eliminate anomalies in the transmission of modulated line spectra.This research intends to offer technical support and a theoretical foundation for reducing target vibrations and noise, as well as extracting weak signals of high-order line spectra in the far field.


are initial phase.The amplitude modulation signal exhibits a ratio of 1

Figure 4 .Figure 5 .Figure 6 .
Figure 4.The abnormality of the transimission loss in the free field