Mathematical Models and Numerical Methods for High Frequency Waves

The numerical approximation of high frequency wave propagation is important in many applications.Examples include the simulation of seismic,acoustic,optical waves and microwaves.When the frequency of the waves is high,this is a difficult multiscale problem.The wavelength is short compared to the overall size of the computational domain and direct simulation using the standard wave equations is very expensive.Fortunately,there are computationally much less costly models,that are good approximations of many wave equations precisely for very high frequencies.Even for linear wave equations these models are often nonlinear.The goal of this paper is to review such mathematical models for high frequency waves,and to survey numerical methods used in simulations.We focus on the geometrical optics approximation which describes the infinite frequency limit of wave equations.We will also discuss finite frequency corrections and some other models.

Electromagnetic Resonance of Astigmatic Gaussian Beam to the High Frequency Gravitational Waves

The high frequency gravitational waves （around lOS-lO12 Hz） could interact with a specially designed electro- magnetic resonance system. It is found that the power of transverse perturbative photon flux （PPF） of an electromagnetic resonance system can be improved significantly by virtue of an astigmatic Caussian beam. Cor- respondingly the signal-to-noise ratio （SNR） would also be improved. When the eccentric ratio of waist satisfying w0x ： w0y 〉 1, the peak value of signal photon flux could be raised by 2-4 times with typical systematic parameters, while the background photon flux would be depressed. Therefore, the ratio of transverse PPF to background photon flux （i.e., SNR） can be further improved 3-8 times with dimensionless amplitude of relic gravitational wave ht = 10-36.

Detection of Gravitational Waves with Semi Classical Features and Resulting Cosmological Implications

The author argues in this document that initial vacuum state values possibly responsible for GW generation in relic conditions in the initial onset of inflation may have a temporary unsqueezed, possibly even coherent initial value, which would permit in certain models classical coherent initial gravitational wave states. Furthermore, several arguments pro and con as to if or not initial relic GW should be high frequency will be presented, with the reason given why earlier string models did NOT favor low frequency relic GW from the big bang. What is observed is that large higher dimensions above our 4 Dimensional space time, if recipients of matter-energy from collapse and re birth of the universe are enough to insure low relic GW. The existence of higher dimensions, in itself if the additional dimensions are small and compact will have no capacity to lower the frequency limit values of relic GW, as predicted by Giovannini, et al. in 1995.

Noise in a coupling electromagnetic detecting system for high frequency gravitational waves

This paper discusses the basic categories of noise in detecting high frequency gravitational waves in the microwave band （-0.1-10GHz）, which contain shot noise from the laser and the thermal radiation photons, thermal noise from statistical fluctuation of the thermal photons and fluctuation of the temperature, radiation press noise on the fractal membrane, the noise caused by the scattering of the Gaussian Beam （GB） in the detecting tube and noise in the microwave radiometers. The analysis shows that a reasonable signal-to-noise ratio may be achieved for a detecting device with the fixed power of GB （105 W）, only when the temperature of the environment is no more than T=I K, and the optimal length of the microwave radiometers is about 0.3 m.

Energy momentum pseudo-tensor of high frequency gravitational waves and their dynamical back-reaction

To describe properties of the high frequency gravitational wave （HFGW） propagating through the vacuum gravitational field in Robertson-Walker background space-time, we calculated its energy momentum pseudo-tensor （EMPT） in the limit of short wavelengths by taking the Brill-Hartle average on the second order perturbation of the Einstein tensor over several wavelengths. By rewriting the EMPT as a form of perfect fluid, the dynamical back-reaction of HIFGW on the background spacetime was discussed. The result shows that the energy density of HFGW, which is in the gauge we chose, is positive definite. The HFGW serves as a source for curving the background space-time and affects the dynamical evolution and time evolution of the scale factor of the Robertson-Walker metric.

Wave height measurement in the Taiwan Strait with a portable high frequency surface wave radar

As an important equipment for sea state remote sensing, high frequency surface wave radar （HFSWR） has received more and more attention. The conventional method for wave height inversion is based on the ratio of the integration of the second-order spectral continuum to that of the first-order region, where the strong external noise and the incorrect delineation of the first- and second-order Doppler spectral regions due to spectral aliasing are two major sources of errors in the wave height. To account for these factors, two more indices are introduced to the wave height estimation, i.e., the ratio of the maximum power of the second-or- der continuum to that of the Bragg spectral region （RSCB） and the ratio of the power of the second harmonic peak to that of the Bragg peak （RSHB）. Both indices also have a strong correlation with the underlying wave height. On the basis of all these indices an empirical model is proposed to estimate the wave height. This method has been used in a three-months long experiment of the ocean state measuring and analyzing ra- dar, type S （OSMAR-S）, which is a portable HFSWR with compact cross-loop/monopole receive antennas developed by Wuhan University since 2006. During the experiment in the Taiwan Strait, the significant wave height varied from 0 to 5 m. The significant wave heights estimated by the OSMAR-S correlate well with the data provided by the Oceanweather Inc. for comparison, with a correlation coefficient of 0.74 and a root mean square error （RMSE） of 0.77 m. The proposed method has made an effective improvement to the wave height estimation and thus a further step toward operational use of the OSMAR-S in the wave height extraction.

Surface tidal currents in the open sea area to the east of the Zhoushan Islands measured with high frequency surface wave radar

Based on the quasi-harmonic analysis of 11 d vector ocean currents obtained from two high frequency sur- face wave radars located at Zhujiajian Island and Shengshan Island, the spatial distribution characteristics of surface tidal currents in the open sea area to the east of the Zhoushan Islands of Zhejiang Province, China are studied. The following conclusions are drawn from the analysis： the tidal current pattern in the open sea area to the east of Zhoushan Islands is primarily regular semidiurnal, which is significantly affected by the shallow water constituents. The directions of the major axes of tidal current ellipses of M2 lie approx- imately in the NW-SE direction. With the increasing of distance away from the coast, the directions of the tidal current ellipses gradually shift toward the E-W direction. The tidal currents are mainly reversing cur- rents. The spatial distribution of probable maximum current velocities decreases gradually from northeast to southwest which is basically in accordance with the spatial distribution of measured maximum current velocities. The residual currents near the coast are larger than those far away from the coast. The directions of the residual currents are basically north by east, and the angle to the due north increases gradually with the increasing distance away from the coast. The topography shows a certain impact on the spatial distri- bution of shallow water constituents, the rotation of tidal currents, the probable maximum currents and the residual currents.

Contrast validation test for retrieval method of high frequency ground wave radar

In this paper, on the basis of the working principles of high frequency ground wave radar for retrieval of ocean wave and sea wind elements were used to systematically study the data obtained from contrast validation test in Zhoushan sea area of Zhejiang Province on Oct. 2000, to validate the accuracy of OSMAR2000 for wave and wind parameters, and to analyze the possible error caused when using OSMAR2000 to retrieve ocean parameters.

Experimental Studies on High-Frequency Performance of the Inverse Control Magneto-Rheological Damper

Severe vibration of underground structures may be induced under blast loads. According to the characteristics of the explosion-induced ground shock wave, a new-type damper, inverse control magneto-rheological（MR） damper was designed to control the vibration, The high-frequency performance test of the MR damper was carried out on the small shaking table. It is shown that the performance can be modeled by use of the modified Bouc-Wen model, and the Parameters of the model keep stable in the range of 15--50 Hz.

Signal photon flux generated by high-frequency relic gravitational waves

The power spectrum of primordial tensor perturbations Pt increases rapidly in the high frequency region if the spectral index nt 〉 0. It is shown that the amplitude of relic gravitational waves ht （5×109 Hz） varies from 10-36 to 10-25 while rtt varies from -6.25 × 10-3 to 0.87. A high frequency gravitational wave detector proposed by F,-Y, Li detects gravitational waves through observing the perturbed photon flux that is generated by interaction between relic gravitational waves and electromagnetic field. It is shown that the perturbative photon flux N1x （5 × 109 Hz） varies from 1.40× 10-4 s-i to 2.85× 107 s-i while nt varies from -6.25 ×10-3 to 0.87, Correspondingly, the ratio of the transverse perturbative photon flux N1x to the background photon flux varies from 10-28 to 10-16.

Very high-frequency gravitational waves from magnetars and gamma-ray bursts

Extremely powerful astrophysical electromagnetic（EM） systems could be possible sources of highfrequency gravitational waves（HFGWs）. Here, based on properties of magnetars and gamma-ray bursts（GRBs）, we address ＂Gamma-HFGWs＂（with very high-frequency around 1020 Hz） caused by ultra-strong EM radiation（in the radiation-dominated phase of GRB fireballs） interacting with super-high magnetar surface magnetic fields（～1011 T）.By certain parameters of distance and power, the Gamma-HFGWs would have far field energy density ？gw around10-6, and they would cause perturbed signal EM waves of～10-20 W/m2 in a proposed HFGW detection system based on the EM response to GWs. Specially, Gamma-HFGWs would possess distinctive envelopes with characteristic shapes depending on the particular structures of surface magnetic fields of magnetars, which could be exclusive features helpful to distinguish them from background noise. Results obtained suggest that magnetars could be involved in possible astrophysical EM sources of GWs in the very high-frequency band, and Gamma-HFGWs could be potential targets for observations in the future.

Computation of High Frequency Wave Diffraction by a Half Plane via the Liouville Equation and Geometric Theory of Diffraction

We construct a numerical scheme based on the Liouville equation of geometric optics coupled with the Geometric Theory of Diffraction(GTD)to simulate the high frequency linear waves diffracted by a half plane.We first introduce a condition,based on the GTD theory,at the vertex of the half plane to account for the diffractions,and then build in this condition as well as the reflection boundary condition into the numerical flux of the geometrical optics Liouville equation.Numerical experiments are used to verify the validity and accuracy of this new Eulerian numerical method which is able to capture the moments of high frequency and diffracted waves without fully resolving the high frequency numerically.

Adaptive modified hough transform track initiator forHFSWR tracking of fast and small targets

High frequency surface wave radar (HFSWR) is well proved to have over the horizon (OTH) detection capability to weak aerial targets, such as concealed airplanes or cruise missiles. The most important problem of detection of fast and small targets using HFSWR is earlier warning, i.e. enlargement of detection range oftargets. Therefore, the detection threshold should be decreased as low as possible, but numerous false alarms are brought about at the same time. On this condition, conventional track initiation techniques, which normally require the probability of false alarm to be at the level of 10-6, will initiate enormous false tracks and lead to abnormal operation of tracking system. An adaptive modified hough transform (AMHT) track initiator is proposed accordingly and the relation of detection range to the performance of track initiator is analyzed in this paper. Simulations are performed to confirm the capability of track initiation to fast and small targets in dense clutter by AMHT track initiator. The tolerable probability of false alarm of detector can reach the level of 10 -3 . And it performs better than track initiator based on modified hough transform (MHT).

Application of the Three-Dimensional Shallow Sea and Continental Shelf Mode for Inversion of Undercurrent

Abstract. Due to the incapability of high frequency surface wave radar （HFSWR） to detect undercurrent parameters, a new algorithm is proposed to apply a three-dimensional （3D） nonlinear barotropic shallow sea and continental shelf model in coordinate system to the inversion of undercurrent. The calculation domain of this model is the area detected by HFSWR. Considering the benthal topography of the detected area and the ocean dynamic parameters, such as surface current, wind and wave detected by HFSWR, the relation between surface current and undercurrent is established in this model, Accordingly, the undercurrent parameters of corresponding area are obtained. The inversion results agree with the law of ocean dynamics and reproduce the movement of undercurrent.

Effective Sea Clutter Spectrum Extraction Method for HFSWR in Adverse Conditions

An effective approach in solving the sea clutter spectrum extraction problem is studied in the paper.Different from the conventional signal to noise ratio（SNR）method based on Doppler frequency or range domain information,a method is developed to characterize the differences between the sea echo and those interferences are by signal to interference plus noise ratio（SINR）which jointly utilizing the range,Doppler frequency and azimuth domain information.Furthermore,these differences can be adaptable to adverse conditions by forming the necessary boundaries and constraints in searching of the maximum SINR,which greatly promotes the extraction of sea clutter spectrum.The real high frequency surface wave radar（HFSWR）data demonstrate that the proposed method is less influenced by those interferences and can effectively extract the sea clutter spectrum even under the adverse conditions.Furthermore,it has been shown as an effective method for ship detection and sea state remote sensing of HFSWR.

Spatial Distribution Characteristics of Surface Tidal Currents in the Southwest of Taiwan Strait

This study was conducted on the spatial distribution characteristics of surface tidal currents in the southwestern Taiwan Strait based on the quasi-harmonic analysis of current data obtained by two high frequency surface wave radar(HFSWR) systems. The analysis shows that the tidal current pattern in the southwestern Taiwan Strait is primarily semi-diurnal and influenced significantly by shallow water constituents. The spatial distribution of tidal current ellipses of M2 is probably affected by the interaction between two different systems of tide wave, one from the northern mouth of Taiwan Strait and the other from the Bashi Channel. The directions of the major axes of M2 tidal current ellipses coincide roughly with the axis of the Taiwan Strait. The spatial distribution of the magnitudes of the probable maximum current velocity(PMCS) shows gradual increase of the velocity from northeast to southwest, which is in accordance with the spatial distribution of the measured maximum current velocity(MMCS). The directions of the residual currents are in accordance with the direction of the prevailing monsoon wind at the Taiwan Strait and the direction of the Taiwan warm current during summer. The bathymetry also shows a significant effect on the spatial distribution characteristics of tidal currents.

First order sea clutter cross section for bistatic shipborne HFSWR

This paper studies the development on the first order sea clutter cross section for bistatic high frequency surface wave radar (HFSWR). Based on the received first order electric field expression, a closed-form of cross sections is derived to account for the case of receiving antenna array being mounted on the shipborne platform. The uniform linear motion and sway motion components are assumed to be responsible for the observed differences in comparison with the bistatic fixed antenna case. Correspondingly, simulations are conducted to study the sea clutter spectral characteristics for these two cases versus different system parameters and sea state conditions. It is shown numerically that the forward motion component will spread the Bragg lines severely and the influence triggered by the sway motion can be explained as the Bessel function modulation of the ordinary sea clutter spectra. The obtained results have important implications in the application of shipborne HFSWR technology to ocean remote sensing and target detection.

Multi-Valued Solution and Level Set Methods in Computational High FrequencyWave Propagation

We review the level set methods for computing multi-valued solutions to a class of nonlinear first order partial differential equations,including Hamilton-Jacobi equations,quasi-linear hyperbolic equations,and conservative transport equations with multi-valued transport speeds.The multivalued solutions are embedded as the zeros of a set of scalar functions that solve the initial value problems of a time dependent partial differential equation in an augmented space.We discuss the essential ideas behind the techniques,the coupling of these techniques to the projection of the interaction of zero level sets and a collection of applications including the computation of the semiclassical limit for Schr¨odinger equations and the high frequency geometrical optics limits of linear wave equations.

Iterative physical optics model for electromagnetic scattering and Doppler analysis

An iterative physical optics（IPO） model is proposed to solve extra large scale electric electromagnetic（EM） scattering from randomly rough surfaces. In order to accelerate the convergence of the IPO model, the forward-backward methodology and its modification with underrelaxation iteration are developed to simulate the rough surface scattering; the local iteration methodology and the fast far field approximation（Fa FFA） in the matrix-vector product are proposed to reduce greatly the computational complexity. These techniques make Monte Carlo simulations possible. Thus, the average Doppler spectra of backscattered signals obtained from the simulations are compared for different incident angles and sea states. In particular, the simulations show a broadening of the Doppler spectra for a more complicated sea state at a low grazing angle（LGA）.

A TAILORED FINITE POINT METHOD FOR THE HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS IN HETEROGENEOUS MEDIUM

In this paper, we propose a tailored-finite-point method for the numerical simulation of the Helmholtz equation with high wave numbers in heterogeneous medium. Our finite point method has been tailored to some particular properties of the problem, which allows us to obtain approximate solutions with the same behaviors as that of the exact solution very naturally. Especially, when the coefficients are piecewise constant, we can get the exact solution with only one point in each subdomain. Our finite-point method has uniformly convergent rate with respect to wave number k in L^2-norm.