Normal Mode Theory Research Articles

Segmented propagation model for the paths and travel times of surface duct leaky signals.

In the deep ocean environment with a surface duct, sound propagating within the duct leaks into the geometric shadow zone below it. However, the propagation paths and time of these leaky parts have not been fully characterized. This paper investigates the mechanism of surface duct leaky (SDL) signals based on normal mode theory. It reveals that SDL signals are caused by specific modes with grazing angles close to zero at the bottom of the surface duct. Combining the theory of diffracted sound rays, the study proposes a Segmented Propagation model (SPM) for SDL signals. The propagation paths of SDL signals are divided into three segments: S1, which extends from the source to the surface duct; S2, the segment propagating within the surface duct; and S3, the segment leading from the surface duct to the receiver. The proposed SPM describes the propagation mechanism of SDL signals and allows for precise calculation of their propagation time. Experimental data from the western Pacific are used to verify the SPM.

On the temperature dependence of the density of states of liquids at low energies

We report neutron-scattering measurements of the density of states (DOS) of water and liquid Fomblin in a wide range of temperatures. In the liquid phase, we confirm the presence of a universal low-energy linear scaling of the experimental DOS as a function of the frequency, g(ω)=a(T)ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g(\\omega )= a(T) \\omega $$\\end{document}, which persists at all temperatures. The low-frequency scaling of the DOS exhibits a sharp jump at the melting point of water, below which the standard Debye’s law, g(ω)∝ω2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g(\\omega ) \\propto \\omega ^2$$\\end{document}, is recovered. On the contrary, in Fomblin, we observe a continuous transition between the two exponents reflecting its glassy dynamics, which is confirmed by structure measurements. More importantly, in both systems, we find that the slope a(T) grows with temperature following an exponential Arrhenius-like form, a(T)∝exp(-⟨E⟩/T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a(T) \\propto \\exp (-\\langle E \\rangle /T)$$\\end{document}. We confirm this experimental trend using molecular dynamics simulations and show that the prediction of instantaneous normal mode (INM) theory for a(T) is in qualitative agreement with the experimental data.

Novel Criteria to Provide a Locality/Normality Degree in Molecules and Their Relevance in Physical Chemistry.

In contrast to the traditional analysis of molecules using local mode behavior, where the degree of locality is given through a function in terms of Morse potential parameters, new criteria for locality/normality (LN) suitable for application to any molecular system are proposed. The approach is based on analysis of the connection between the algebraic normal and local mode representations. It is shown that both descriptions are equivalent as long as the polyad (total number of quanta) in the local representation is not conserved. The constraint of a local polyad conservation naturally provides a criterion for assigning an LN degree in quantitative form, without an analogue in configuration space. The correlation between the different parameters reveals the physical properties of molecules. A clear connection between the LN degree (based on the fundamentals) and spectroscopic properties is also presented, suggesting a promising approach for identifying mixtures of isotopologues.

A novel model order reduction technique for solving horizontal refraction equations in the modeling of three-dimensional underwater acoustic propagation

Modeling three-dimensional (3D) underwater acoustic propagation is vital for underwater detection, localization, and navigation. This article introduces a novel model order reduction (MOR) technique to solve horizontal refraction equations (HREs) in the modeling of 3D underwater acoustic propagation. This approach relies on an adiabatic approximation of the 3D sound field, representing the field as a combination of local vertical modes with their modal coefficients governed by a system of two-dimensional (2D) HREs. Inspired by normal mode theory, the coefficients in the expansion over vertical modes are determined by projecting them onto a lower-dimensional, orthogonal space defined by their transverse eigenfunctions. By artificially truncating the horizontal domain in the transverse directions using two perfectly matched layers (PMLs), the eigenproblem associated with the transverse eigenfunctions of the modal coefficients is closed and thus can be solved through a modal projection method. The modal projection method enables fast computation of modal coefficients in a longitudinally invariant environment within seconds, offering a naturally wide-angle solution covering ±90°. The MOR method is extended to encompass fully 3D cases by introducing an admittance matrix, a memory-saving strategy that prevents numerical overflow when the longitudinal range is large. Moreover, the fact that the outer boundaries of the PMLs are range-independent allows the proposed MOR technique to perform well on a coarse grid when employing the Magnus scheme, significantly saving the numerical cost for the fully 3D simulation. Numerical simulations are provided for both longitudinally invariant and fully 3D scenarios, demonstrating the high accuracy and efficiency of the proposed MOR technique in solving HREs.

Nonlinear Normal Modes of Vibrating Mechanical Systems: 10 Years of Progress

Abstract This paper contains a review of the theory and applications of nonlinear normal modes, which have been developed during last decade. This review has more than 200 references. It is a continuation of two previous review papers by the same authors. The following theoretical issues of nonlinear normal modes are treated: basic concepts and definitions; application of the normal form theory for nonlinear modes construction; nonlinear modes in finite degrees-of-freedom systems; resonances and bifurcations; reduced-order modeling; nonlinear modes in stochastic dynamical systems; numerical methods; identification of mechanical systems using nonlinear modes. The following applied issues of this theory are treated in this review: experimental measurement of nonlinear modes; nonlinear modes in continuous systems; engineering applications (aerospace engineering, power engineering, piecewise-linear systems and structures with dry friction); nonlinear modes in nanostructures and physical systems; targeted energy transfer and absorption problem.

Subarray-based joint source localization in shallow water waveguide via subspace intersection

By the use of an underwater bottom-mounted Horizontal Linear Array (HLA), the two-dimensional joint localization technique and its accuracy are analyzed in this article. The proposed method takes advantage of arrival angles measured by a dual-subarray system consisting of a large-aperture HLA. Then, for the system, expressions for source position estimate and error variation are derived. Analysis of the Geometrical Dilution of Precision, which is a measure of localization error defined in Eq. (4), reveals that the localization error is mainly related to the source bearing, subarray interval and their measurement errors. In order to investigate the impact of the underwater waveguide, numerical simulations were carried out based on the normal mode theory. By using the distribution of acoustic intensity and structure of modal arrival, it is proved that the localization accuracy increases as the grazing angle and the subarray interval increase. However, when the measurement errors of source bearing and subarray interval increase, the localization accuracy will decrease. In addition, the obtained bearing estimation result provides an evident characteristic of beam spreading. When sources are near the end-fire direction of the dual-subarray system, significant bearing estimation error occurs, the negative effect of which is then presented. By applying the Subspace Intersection method, unbiased bearing estimation has been achieved, with relatively reduced computational complexity. Most of all, the improved localization performance is validated by experimental data collected in the South China Sea.

Fluid-Dynamic Mechanisms Underlying Wind Turbine Wake Control with Strouhal-Timed Actuation

A reduction in wake effects in large wind farms through wake-aware control has considerable potential to improve farm efficiency. This work examines the success of several emerging, empirically derived control methods that modify wind turbine wakes (i.e., the pulse method, helix method, and related methods) based on Strouhal numbers on the O(0.3). Drawing on previous work in the literature for jet and bluff-body flows, the analyses leverage the normal-mode representation of wake instabilities to characterize the large-scale wake meandering observed in actuated wakes. Idealized large-eddy simulations (LES) using an actuator-line representation of the turbine blades indicate that the n=0 and ±1 modes, which correspond to the pulse and helix forcing strategies, respectively, have faster initial growth rates than higher-order modes, suggesting these lower-order modes are more appropriate for wake control. Exciting these lower-order modes with periodic pitching of the blades produces increased modal growth, higher entrainment into the wake, and faster wake recovery. Modal energy gain and the entrainment rate both increase with streamwise distance from the rotor until the intermediate wake. This suggests that the wake meandering dynamics, which share close ties with the relatively well-characterized meandering dynamics in jet and bluff-body flows, are an essential component of the success of wind turbine wake control methods. A spatial linear stability analysis is also performed on the wake flows and yields insights on the modal evolution. In the context of the normal-mode representation of wake instabilities, these findings represent the first literature examining the characteristics of the wake meandering stemming from intentional Strouhal-timed wake actuation, and they help guide the ongoing work to understand the fluid-dynamic origins of the success of the pulse, helix, and related methods.

The waveguide invariant close to the deep-water bottom

Waveguide invariant, denoted by β, describes the dispersive nature of the underwater acoustic waveguide. In most range-independent shallow-water waveguides β≈1, whereas β is no longer constant in deep water and is more appropriately described by a distribution. This paper investigates the characteristic of β close to the deep-water bottom. The adiabatic normal mode theory is used to analyze β resulting from the bottom-bounce dominated acoustic field qualitatively. Then, the Wentzel–Kramers–Brillouin approximation with bottom reflection phase shift and the dominance of the acoustic field by groups of modes are employed to amend the original waveguide invariant theory. The interference between the phase-compensated dominant modal groups is calculated to obtain the value of β and analyze the distribution in the two-dimensional spectrum of the interference pattern. For a typical deep-water waveguide, both qualitative and quantitative analyses indicate that β≈1 when close to the bottom instead of a distribution. Simulation and experiment data demonstrate that the interference between phase-compensated dominant modal groups produces particular structure in the two-dimensional spectrum resulting from β being approximately equal to 1. The characteristic of β proposed in this paper has potential application in remote ranging the underwater acoustic source in deep water, which is verified using experimental data.

Analysis of line spectral fluctuation characteristics for source depth discrimination in shallow water

The difference in intensity fluctuation between a surface signal and a submerged signal can be used to discriminate the source depth in the shallow water waveguide. However, the properties of fluctuation distinction in intensity between surface source and submerge source are rarely studied, resulting in the poor performance of fluctuation-based methods sometimes. In this work, the intensity fluctuations caused by source depth fluctuations is analyzed by differential method and variance statistics method to figure out the physics of intensity fluctuations, and to present the suitable quantity for depth discrimination. Firstly, the derivative expression of real part, power and amplitude of the pressure field are respectively derived from the normal mode theory, hence their fluctuation quantities are specified. Then, the numerical examples, including the factor of receiver depth, source frequency and source range, are treated to verify the derivation and summarize the characteristic of fluctuations. The property of the intensity fluctuations is also compared with the distribution of normalized quantity on the depth dimension. Finally, the SWellEx-96 experimental data processing results validate the conclusions. The processing results of derivation, simulation data and experimental data show that the interactions between modes excited by the source give rise to the fluctuations. Meanwhile, the property of the interactions is affected by the received depth, source range and source frequency. Amplitude is capable of building a quantity based on fluctuations for source depth discrimination. For surface and submerged sources with lower frequency, their distinction in their fluctuations is stably represented by amplitude. In the SWellEx-96 experiment, the normalized values calculated from the amplitude of surface noise and submerged sound sources show the lowest difference of 0.5 and the highest difference of 2.5, indicating the effectiveness of using amplitude fluctuations to discriminate the sound source depth.

Heterogeneous-elasticity theory of instantaneous normal modes in liquids

Since decades, the concept of vibrational density of states in glasses has been mirrored in liquids by the instantaneous-normal-mode spectrum. In glasses instantaneous configurations are believed to be situated close to minima of the potential-energy hypersurface and all eigenvalues of the associated Hessian matrix are positive. In liquids this is no longer true, and modes corresponding to both positive and negative eigenvalues exist. The instantaneous-normal-mode spectrum has been numerically investigated in the past, and it has been demonstrated to bring important information on the liquid dynamics and transport properties. A systematic deeper theoretical understanding is now needed. Heterogeneous-elasticity theory has proven to be particularly successful in explaining many details of the low-frequency excitations in glasses, ranging from the thoroughly studied boson peak, to other anomalies related to the crossover between wave-like and random-matrix-like excitations. Here we present an extension of heterogeneous-elasticity theory to the liquid state, and show that the outcome of the theory agrees well to the results of extensive molecular-dynamics simulations of a model liquid at different temperatures. We find that the spectrum of eigenvalues ρ(λ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho (\\lambda )$$\\end{document} has a sharp maximum close to (but not at) λ=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda =0$$\\end{document}, and decreases monotonically with |λ|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\lambda |$$\\end{document} on both its stable and unstable side. We show that the spectral shape strongly depends on temperature, being symmetric at high temperatures and becoming rather asymmetric at low temperatures, close to the dynamical critical temperature. Most importantly, we demonstrate that the theory naturally reproduces a surprising phenomenon, a zero-energy spectral singularity with a cusp-like character developing in the vibrational spectra upon cooling. This feature, known from a few previous numerical studies, has been generally overlooked in the past due to a misleading representation of the data. We provide a thorough analysis of this issue, based on both very accurate predictions of our theory, and computational studies of model liquid systems with extended size.

Analysis and application of spatio-temporal correlation characteristics of measured ocean ambient noise

Gain in sonar signal processing is closely linked to the spatio-temporal coherence characteristics of the ocean ambient noise field. The gain relies on the spatio-temporal coherence coefficient of the signal and noise, with the aim of maximizing the former and minimizing the latter. Currently, there has been ample research on the spatial characteristics of the noise field, while studies on the temporal coherence characteristics of single-point noise fields are limited. In this paper, a temporal and spatial coherence model of wind-induced noise is established based on the normal mode theory, and the impact of bandwidth and spectral structure on the time coherence coefficient of wind-induced noise is analyzed.To measure the environmental noise in 103 m of shallow water in the South China Sea, a 64-element vertical array was utilized. The results demonstrate that the measured vertical spatial correlation coefficient aligns closely with the theoretical findings when wind-induced noise is predominant. The measured temporal correlation coefficient of narrowband noise exhibits a cosine function with gradual attenuation of the envelope, and the correlation radius remains independent of frequency. As the bandwidth increases, the measured broadband noise temporal correlation radius gradually decreases. Wind speed has minimal influence on the temporal and spatial correlation coefficients of wind-induced noise. In the presence of interferences such as ships and pulses, the measured temporal and spatial correlation coefficients of wind-induced noise exhibit larger values in certain intervals.Lastly, leveraging the characteristic that the temporal coherence radius of noise is significantly smaller than that of the line spectrum signal, a single-hydrophone cross-spectrum method is proposed to process the line spectrum signal. This approach effectively enhances the signal-to-noise ratio of the line spectrum.

Coupled-mode separation of multiply scattered wavefield components in two-dimensional waveguides.

For a waveguide that is invariant in one of the horizontal directions, this paper presents a mathematically exact partial-wave decomposition of the wavefield for assessment of multiple scattering by horizontally displaced medium anomalies. The decomposition is based on discrete coupled-mode theory and combination of reflection/transmission matrices. In particular, there is no high-frequency ray approximation. Full details are presented for the scalar case with plane-wave incidence from below. An application of interest concerns ground motion induced by seismic waves, which may be severely amplified by local medium anomalies such as alluvial valleys. Global optimization techniques are used to design an artificial medium termination at depth for a normal-mode representation of the field. A derivation of a horizontal source array to produce an incident plane wave gives, as a by-product, an extension of a previous Fourier-transform relation involving Bessel functions. Like purely numerical methods, such as finite differences and finite elements, the method can handle all kinds of (two-dimensional) anomaly shapes.

Broadband Acoustic Field in a Shallow-Water Waveguide with an Inhomogeneous Bottom

A broadband (35–1000 Hz) sound field formed by a point source in a shelf zone with inhomogeneous bottom sediment structure is studied using numerical modeling. The shelf depth is about 30 m and the maximum distance is 10 km. The transitional zone from the bottom with a sound speed of 1400 m/s to the bottom with a speed of 1600 m/s is chosen as the model inhomogeneity. The normal mode theory and wide-angle parabolic equations are used for sound field calculations. Numerical experiments show that the manifestation of horizontal refraction is noticeable at low frequencies (below 100 Hz). It leads to an increase in the amplitude of the low-frequency sound pulse propagating along the transitional zone by more than 10 dB in comparison with a similar waveguide with a homogeneous bottom. At frequencies above 100 Hz, the dominant effect is the mode coupling, causing the appearance of quasi-periodic oscillations of modal amplitude in the frequency domain. The conclusions from the simplified model are confirmed by calculations for the real structure of bottom sediments in the Kara Sea.

Singular Points Location Pattern Analysis based on Modes Interference

<p>Given the lack of research on the regularity of distribution of singular points, the singular points under different influence factors are analyzed. The location of singular points for adjacent modes group (AMG) generated by the interference modes is derived using normal modes theory in the far field of the ideal shallow waveguide. The singular points are verified by simulation, which compares the distribution of the acoustic field and singular points. Moreover, it is concluded that the number and distribution of singular points in the period of the interference structure of the field (PISF) has a relationship with order (lower order of AMG) and adjacent order (difference between two orders of AMG). The arrangement of upper-lower-two-points increases by 2 with each increase of order and is distributed at both ends of PISF. The arrangement of rhombic-four-points increases by 1 with each increase of adjacent order and increases alternately at both ends of PISF. Meanwhile, there is a periodic relationship between the depth of the source and the location of singular points.</p> <p> </p>

Parameter uncertainty and noise effects on the global dynamics of an electrically actuated microarch

This work aims to study the effect of uncertainties and noise on the nonlinear global dynamics of a micro-electro-mechanical arch obtained from an imperfect microbeam under an axial load and electric excitation. An adaptative phase-space discretization strategy based on an operator approach is proposed. The Ulam method, a classical discretization of flows in phase-space, is extended here to nondeterministic cases. A unified description is formulated based on the Perron–Frobenius, Koopman, and Foias linear operators. Also, a procedure to obtain global structures in the mean sense of systems with parametric uncertainties is presented. The stochastic basins of attraction and attractors’ distributions replace the usual basin and attractor concepts. For parameter uncertainty cases, the phase-space is augmented with the corresponding probability space. The microarch is assumed to be shallow and modeled using a nonlinear Bernoulli–Euler beam theory and is discretized by the Galerkin method using as interpolating function the linear vibration modes. Then, from the discretized multi degree of freedom model model, an accurate single degree of freedom reduced order model, based on theory of nonlinear normal modes, is derived. Several competing attractors are observed, leading to different (acceptable or unacceptable) behaviors. Extensive numerical simulations are performed to investigate the effect of noise and uncertainties on the coexisting basins of attraction, attractors’ distributions, and basins boundaries. The appearance and disappearance of attractors and stochastic bifurcation are observed, and the time-dependency of stochastic responses is demonstrated, with long-transients influencing global behavior. To consider uncertainties and noise in design, a dynamic integrity measure is proposed via curves of constant probability, which give quantitative information about the changes in structural safety. For each attractor, the basin robustness as a function of a stochastic parameter is investigated. The weighted basin area can quantify the integrity of nondeterministic cases, being also the most natural generalization of the global integrity measure. While referring to particular micro-electro-mechanical system, the relevance of the dynamical integrity analysis for stochastic systems to quantify tolerances and safety margins is underlined here.

Model order reduction for seismic waveform modelling: inspiration from normal modes

SUMMARY The computational cost of full waveform simulation in seismological contexts is known to be expensive and generally requires large clusters of computers working in parallel. Although there have been many methods proposed over recent years to reduce this burden, in this work, we focus on a particular method called model order reduction (MOR) whereby a full waveform system of equations is projected onto a lower dimensional space to reduce computational and memory requirements at the cost of introducing approximation errors. In this paper, inspired by normal mode (NM) theory, we use the eigenmodes of the seismic wave equation to span this lower dimensional space. From this we argue that NM theory can be seen as an early form of MOR. Using this as inspiration, we demonstrate how free body oscillations and a form of Petrov–Galerkin projection can be applied in regional scale problems utilizing recent advanced eigensolvers to create a MOR scheme. We also demonstrate how this can be applied to inverse problems. We further conjecture that MOR will have an important role to play in future full waveform applications, particularly those of a time-critical nature such as seismic hazard monitoring.

Source estimation of the tsunami later phases associated with the 2022 Hunga Tonga volcanic eruption

SUMMARY On 15 January 2022, a large eruption of the Hunga Tonga-Hunga Ha’apai volcano in Tonga triggered globally observed tsunami waves. While the first arrival in the observed wave trains is now widely known to be related to the atmospheric Lamb wave generated by the eruption, large later phases, whose amplitudes were comparable to the first ones, were also recorded. In this study, we estimated the source of the later phases based on the Vespa analysis and proposed a new numerical scheme to reproduce them. The Vespa analysis estimates the arrival time and incident angle of each signal by a slant-stack process using its theoretical traveltime. The Vespa analysis revealed that small atmospheric waves excited the large later tsunamis. For the numerical experiments, we used two types of synthetic methods: finite difference method and normal mode theory. We found that both a good atmospheric wave model and bathymetric effect were important to generate the atmospheric-induced tsunamis corresponding to the later phases. A hybrid method calculating tsunamis by the finite difference method with the atmospheric waves by the normal mode theory as the input successfully reproduced the observed records, particularly in amplitude over the entire records.

Observation and analysis of ν1 + ν3 and ν1 + 2ν4 bands of CF4 molecule

Infrared spectra of CF4 molecule were measured for three different temperatures in the range 2160–2210 cm−1 using Fourier transform spectrometer. Non-empirical effective Hamiltonian obtained from the potential energy surface using the contact transformation method was used for the modeling of line positions. The origins of two bands ν1 + ν3 (F2) and ν1 + 2ν4 (F2) at 2186.97 and 2168.21 cm−1 were experimentally determined for the first time. The experimental origin of the weak ν1 + 2ν4 (E) sub-band was tentatively evaluated near 2169.84 cm−1. Ab initio line intensities were calculated from high order dipole moment surface by variational method using the TENSOR computational code in the normal mode representation. The analyses of experimental spectra were carried out using the MIRS program suite accounting for the full tetrahedral symmetry of the molecule.

Machine-learning-based simultaneous detection and ranging of impulsive baleen whale vocalizations using a single hydrophone.

The low-frequency impulsive gunshot vocalizations of baleen whales exhibit dispersive propagation in shallow-water channels which is well-modeled by normal mode theory. Typically, underwater acoustic source range estimation requires multiple time-synchronized hydrophone arrays which can be difficult and expensive to achieve. However, single-hydrophone modal dispersion has been used to range baleen whale vocalizations and estimate shallow-water geoacoustic properties. Although convenient when compared to sensor arrays, these algorithms require preliminary signal detection and human labor to estimate the modal dispersion. In this paper, we apply a temporal convolutional network (TCN) to spectrograms from single-hydrophone acoustic data for simultaneous gunshot detection and ranging. The TCN learns ranging and detection jointly using gunshots simulated across multiple environments and ranges along with experimental noise. The synthetic data are informed by only the water column depth, sound speed, and density of the experimental environment, while other parameters span empirically observed bounds. The method is experimentally verified on North Pacific right whale gunshot data collected in the Bering Sea. To do so, 50 dispersive gunshots were manually ranged using the state-of-the-art time-warping inversion method. The TCN detected these gunshots among 50 noise-only examples with high precision and estimated ranges which closely matched those of the physics-based approach.

Method for measuring the low-frequency sound power from a complex sound source based on sound-field correction in a non-anechoic tank

Similar to air reverberation chambers, non-anechoic water tanks are important acoustic measurement devices that can be used to measure the sound power radiated from complex underwater sound sources using diffusion field theory. However, the problem of the poor applicability of low-frequency measurements in these tanks has not yet been solved. Therefore, we propose a low-frequency acoustic measurement method based on sound-field correction (SFC) in an enclosed space that effectively solves the problem of measuring the sound power from complex sound sources below the Schroeder cutoff frequency in a non-anechoic tank. Using normal mode theory, the transfer relationship between the mean-square sound pressure in an underwater enclosed space and the free-field sound power of the sound source is established, and this is regarded as a correction term for the sound field between this enclosed space and the free field. This correction term can be obtained based on previous measurements of a known sound source. This term can then be used to correct the mean-square sound pressure excited by any sound source to be tested in this enclosed space and equivalently obtain its free-field sound power. Experiments were carried out in a non-anechoic water tank (9.0 m × 3.1 m × 1.7 m) to confirm the validity of the SFC method. Through measurements with a spherical sound source (whose free-field radiation characteristics are known), the correction term of the sound field between this water tank and the free field was obtained. On this basis, the sound power radiated from a cylindrical shell model under the action of mechanical excitation was measured. The measurement results were found to have a maximum deviation of 2.9 dB from the free-field results. These results show that the SFC method has good applicability in the frequency band above the first-order resonant frequency in a non-anechoic tank. This greatly expands the potential low-frequency applications of non-anechoic tanks.