A Wentzel-Kramers-Brillouin(WKB)method is introduced for obtaining a uniform asymptotic solution for underwater sound propagation at very low frequencies in deep ocean.The method utilizes a mode sum and employs the re...A Wentzel-Kramers-Brillouin(WKB)method is introduced for obtaining a uniform asymptotic solution for underwater sound propagation at very low frequencies in deep ocean.The method utilizes a mode sum and employs the reference functions method to describe the solution to the depth-separated wave equation approximately using parabolic cylinder functions.The conditions for the validity of this approximation are also discussed.Furthermore,a formula that incorporates waveguide effects for the modal group velocity is derived,revealing that boundary effects at very low frequencies can have a significant impact on the propagation characteristics of even low-order normal modes.The present method not only offers improved accuracy compared to the classical WKB approximation and the uniform asymptotic approximation based on Airy functions,but also provides a wider range of depth applicability.Additionally,this method exhibits strong agreement with numerical methods and offers valuable physical insights.Finally,the method is applied to the study of very-low-frequency sound propagation in the South China Sea,leading to sound transmission loss predictions that closely align with experimental observations.展开更多
The notes here presented are of the modifications introduced in the application of WKB method.Theproblems of two-and three-dimensional harmonic oscillator potential are revisited by WKB and the new formulationof quant...The notes here presented are of the modifications introduced in the application of WKB method.Theproblems of two-and three-dimensional harmonic oscillator potential are revisited by WKB and the new formulationof quantization rule respectively.It is found that the energy spectrum of the radial harmonic oscillator,which isreproduced exactly by the standard WKB method with the Langer modification,is also reproduced exactly without theLanger modification via the new quantization rule approach.An alternative way to obtain the non-integral Maslov indexfor three-dimensional harmonic oscillator is proposed.展开更多
In this paper, we combine the perturbation method in supersymmetric quantum mechanics with the WKB method to restudy an angular equation coming from the wave equations for a Sehwarzschild black hole with a straight st...In this paper, we combine the perturbation method in supersymmetric quantum mechanics with the WKB method to restudy an angular equation coming from the wave equations for a Sehwarzschild black hole with a straight string passing through it. This angular equation serves as a naive model for our investigation of the combination of supersymmetric quantum mechanics and the WKB method, and will provide valuable insight for our further study of the WKB approximation in real problems, like the one in spheroidal equations, etc.展开更多
The mild-slope equation derived by Berkhoff (1972), has widely been used in the numerical calculation of refraction and diffraction of regular waves. However, it is well known that the random sea waves has a significa...The mild-slope equation derived by Berkhoff (1972), has widely been used in the numerical calculation of refraction and diffraction of regular waves. However, it is well known that the random sea waves has a significant effect in the refraction and diffraction problems. In this paper, a new form of time-dependent mild slope equation for irregular waves was derived with Fade approximation and Kubo's time series concept. The equation was simplified using WKB method, and simple and practical irregular mild slope equation was obtained. Results of numerical calculations are compared with those of laboratory experiments.展开更多
We investigate computationally the attenuation and reflection of Terahertz （THz） wave using targets coated with plasmas. The simulators are the Wentzel-Kramer-Brillouin （WKB） method and finite-difference timedoma...We investigate computationally the attenuation and reflection of Terahertz （THz） wave using targets coated with plasmas. The simulators are the Wentzel-Kramer-Brillouin （WKB） method and finite-difference timedomain （FDTD） method. The relation between the frequency of the incident electromagnetic （EM） wave and the attenuation caused by unmagnitized plasma is analyzed. The results demonstrate that the amount of absorbed power is a decreasing function of the EM wave frequency and the plasma collision frequency. For THz band incident wave, the attenuation that is caused by plasma is small when the plasma has common density and the collision frequency. This conclusion has fine applying foreground for plasma anti stealth.展开更多
Theoretical analysis and numerical calculations of Love wave propagation in layered graded composites with imperfectly bonded interface are described in this paper. On the basis of WKB method, the approximate analytic...Theoretical analysis and numerical calculations of Love wave propagation in layered graded composites with imperfectly bonded interface are described in this paper. On the basis of WKB method, the approximate analytic solutions for Love waves are obtained. By the interface shear spring model, the dispersion relations for Love waves in layered graded composite structures with rigid, slip, and imperfectly bonded interfaces are given, and the effects of the interface conditions on the phase velocities of Love waves in SiC/Al layered graded composites are discussed. Numerical analysis shows that the phase velocity decreases when the defined flexibility parameter is greater. For the general imperfectly bonded interface, the phase velocity changes in the range of the velocities for the rigid and slip interface conditions.展开更多
We present a relation between the Mathieu equation and a particular elliptic curve. We find that the Floquet exponent of the Mathieu equation, for both q《1 and q》1, can be obtained from the integral of a differentia...We present a relation between the Mathieu equation and a particular elliptic curve. We find that the Floquet exponent of the Mathieu equation, for both q《1 and q》1, can be obtained from the integral of a differential one form along the two homology cycles of the elliptic curve. Certain higher order differential operators are needed to generate the WKB expansion. We make a few conjectures about the general structure of these differential operators.展开更多
We study the WKB dispersion equation in non-uniform optical waveguide.There are three methods given in this paper:(1) method of Airy function;(2)method of connection formula;and(3) method of phase shift.At last we mak...We study the WKB dispersion equation in non-uniform optical waveguide.There are three methods given in this paper:(1) method of Airy function;(2)method of connection formula;and(3) method of phase shift.At last we make some remarks.展开更多
Free longitudinal vibrations of non-uniform rods are investigated by a proposed method, which results in a series solution. In a special case, with the proposed method an exact solution with a concise form can be obta...Free longitudinal vibrations of non-uniform rods are investigated by a proposed method, which results in a series solution. In a special case, with the proposed method an exact solution with a concise form can be obtained, which imply four types of profiles with variation in geometry or material properties. However, the WKB (Wentzel-Kramers-Brillouin) method leads to a series solution, which is a Taylor expansion of the results of the proposed method. For the arbitrary non-uniform rods, the comparison indicates that the WKB method is simpler, but the convergent speed of the series solution resulting from the pro-posed method is faster than that of the WKB method, which is also validated numerically using an exact solution of a kind of non-uniform rods with Kummer functions.展开更多
Because the nonlinearity of actual physical processes can be expressed more precisely by the introduction of a non- linear term, the weakly nonlinear Prandtl model is one of the most effective ways to describe the pur...Because the nonlinearity of actual physical processes can be expressed more precisely by the introduction of a non- linear term, the weakly nonlinear Prandtl model is one of the most effective ways to describe the pure katabatic flow （no backgrotmd flow）. Features of the weak nonlinearity are reflected by two factors： the small parameter c and the gradually varying eddy thermal conductivity. This paper first shows how to apply the Wentzel-Kramers-Brillouin （WKB） method for the approximate solution of the weakly nonlinear Prandtl model, and then describes the retrieval of gradually varying eddy thermal conductivity from observed wind speed and perturbed potential temperature. Gradually varying eddy thermal conductivity is generally derived from an empirical parameterization scheme. We utilize wind speed and potential temperature measurements, along with the variational assimilation technique, to de- rive this parameter. The objective function is constructed by the square of the differences between the observation and model value. The new method is validated by numerical experiments with simulated measurements, revealing that the order of the root mean squre error is 10-2 and thus confirming the method＇s robustness. In addition, this me- thod is caoable of anti-interference, as it effectivelv reduces the influence of observation error.展开更多
Since novel optoelectronic devices based on the peculiar behaviors of the tunneling probability, e.g., resonant tunneling devices (RTD) and band-pass filter, are steadily proposed, the analytic transfer matrix (ATM) m...Since novel optoelectronic devices based on the peculiar behaviors of the tunneling probability, e.g., resonant tunneling devices (RTD) and band-pass filter, are steadily proposed, the analytic transfer matrix (ATM) method is extended to study these devices. For several examples, we explore the effect of the scattered subwaves on tunneling; it is shown that the resonant or band-pass structures in tunneling probability are determined by the phase shift results from the scattered subwaves.展开更多
In this paper,by the WKB method the relation between the energy increase of internal inertial gravity waves and heterogeneous atmospheric stratification is derived,and a new generalized wave action is defined and its ...In this paper,by the WKB method the relation between the energy increase of internal inertial gravity waves and heterogeneous atmospheric stratification is derived,and a new generalized wave action is defined and its conservation is proved.展开更多
In this paper the influences of nonuniform stratification on the propagating paths of internal inertial-gravity and pure gravity wave energy are discussed by using the WKB approximation method.The conditions for conse...In this paper the influences of nonuniform stratification on the propagating paths of internal inertial-gravity and pure gravity wave energy are discussed by using the WKB approximation method.The conditions for conservation of wave energy,generalized wave action and wave enstrophy are obtained.The necessary condition of instability for inter- nal gravity waves and the equation governing the refraction of wave rays are derived.Two types of critical levels are giv- en.Finally,the wave rays for different distributions of stratification are calculated by using the fourth-order Runge-Kutta method.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12174048 and 12204128)。
文摘A Wentzel-Kramers-Brillouin(WKB)method is introduced for obtaining a uniform asymptotic solution for underwater sound propagation at very low frequencies in deep ocean.The method utilizes a mode sum and employs the reference functions method to describe the solution to the depth-separated wave equation approximately using parabolic cylinder functions.The conditions for the validity of this approximation are also discussed.Furthermore,a formula that incorporates waveguide effects for the modal group velocity is derived,revealing that boundary effects at very low frequencies can have a significant impact on the propagation characteristics of even low-order normal modes.The present method not only offers improved accuracy compared to the classical WKB approximation and the uniform asymptotic approximation based on Airy functions,but also provides a wider range of depth applicability.Additionally,this method exhibits strong agreement with numerical methods and offers valuable physical insights.Finally,the method is applied to the study of very-low-frequency sound propagation in the South China Sea,leading to sound transmission loss predictions that closely align with experimental observations.
基金National Natural Science Foundation of China under Grant No.10747130the Foundation of East China University of Science and Technology
文摘The notes here presented are of the modifications introduced in the application of WKB method.Theproblems of two-and three-dimensional harmonic oscillator potential are revisited by WKB and the new formulationof quantization rule respectively.It is found that the energy spectrum of the radial harmonic oscillator,which isreproduced exactly by the standard WKB method with the Langer modification,is also reproduced exactly without theLanger modification via the new quantization rule approach.An alternative way to obtain the non-integral Maslov indexfor three-dimensional harmonic oscillator is proposed.
基金supported by the National Natural Science Foundation of China (Grant No. 10875018)
文摘In this paper, we combine the perturbation method in supersymmetric quantum mechanics with the WKB method to restudy an angular equation coming from the wave equations for a Sehwarzschild black hole with a straight string passing through it. This angular equation serves as a naive model for our investigation of the combination of supersymmetric quantum mechanics and the WKB method, and will provide valuable insight for our further study of the WKB approximation in real problems, like the one in spheroidal equations, etc.
基金The research was financially supported by the Doctor degree Program Foundation of State Education Commission of China
文摘The mild-slope equation derived by Berkhoff (1972), has widely been used in the numerical calculation of refraction and diffraction of regular waves. However, it is well known that the random sea waves has a significant effect in the refraction and diffraction problems. In this paper, a new form of time-dependent mild slope equation for irregular waves was derived with Fade approximation and Kubo's time series concept. The equation was simplified using WKB method, and simple and practical irregular mild slope equation was obtained. Results of numerical calculations are compared with those of laboratory experiments.
基金the National Natural Science Foundation of China (60771017)the China Postdoctoral ScienceFoundation (20060390272)
文摘We investigate computationally the attenuation and reflection of Terahertz （THz） wave using targets coated with plasmas. The simulators are the Wentzel-Kramer-Brillouin （WKB） method and finite-difference timedomain （FDTD） method. The relation between the frequency of the incident electromagnetic （EM） wave and the attenuation caused by unmagnitized plasma is analyzed. The results demonstrate that the amount of absorbed power is a decreasing function of the EM wave frequency and the plasma collision frequency. For THz band incident wave, the attenuation that is caused by plasma is small when the plasma has common density and the collision frequency. This conclusion has fine applying foreground for plasma anti stealth.
基金Engineering Research Institute of Peking University (ERIPKU) Joint Building Project of Beijing Education Committee
文摘Theoretical analysis and numerical calculations of Love wave propagation in layered graded composites with imperfectly bonded interface are described in this paper. On the basis of WKB method, the approximate analytic solutions for Love waves are obtained. By the interface shear spring model, the dispersion relations for Love waves in layered graded composite structures with rigid, slip, and imperfectly bonded interfaces are given, and the effects of the interface conditions on the phase velocities of Love waves in SiC/Al layered graded composites are discussed. Numerical analysis shows that the phase velocity decreases when the defined flexibility parameter is greater. For the general imperfectly bonded interface, the phase velocity changes in the range of the velocities for the rigid and slip interface conditions.
基金Supported by the National Natural Science Foundation of China under Grant Nos. 10675061 adn 11175090
文摘We present a relation between the Mathieu equation and a particular elliptic curve. We find that the Floquet exponent of the Mathieu equation, for both q《1 and q》1, can be obtained from the integral of a differential one form along the two homology cycles of the elliptic curve. Certain higher order differential operators are needed to generate the WKB expansion. We make a few conjectures about the general structure of these differential operators.
文摘We study the WKB dispersion equation in non-uniform optical waveguide.There are three methods given in this paper:(1) method of Airy function;(2)method of connection formula;and(3) method of phase shift.At last we make some remarks.
基金supported by the National Natural Science Foundation of China (Grant Nos. 11072157 and 10932006) the Program for Chang-jiang Scholars and Innovative Research Team in University (IRT0971).
文摘Free longitudinal vibrations of non-uniform rods are investigated by a proposed method, which results in a series solution. In a special case, with the proposed method an exact solution with a concise form can be obtained, which imply four types of profiles with variation in geometry or material properties. However, the WKB (Wentzel-Kramers-Brillouin) method leads to a series solution, which is a Taylor expansion of the results of the proposed method. For the arbitrary non-uniform rods, the comparison indicates that the WKB method is simpler, but the convergent speed of the series solution resulting from the pro-posed method is faster than that of the WKB method, which is also validated numerically using an exact solution of a kind of non-uniform rods with Kummer functions.
基金Supported by the National Natural Science Foundation of China(41575026)
文摘Because the nonlinearity of actual physical processes can be expressed more precisely by the introduction of a non- linear term, the weakly nonlinear Prandtl model is one of the most effective ways to describe the pure katabatic flow （no backgrotmd flow）. Features of the weak nonlinearity are reflected by two factors： the small parameter c and the gradually varying eddy thermal conductivity. This paper first shows how to apply the Wentzel-Kramers-Brillouin （WKB） method for the approximate solution of the weakly nonlinear Prandtl model, and then describes the retrieval of gradually varying eddy thermal conductivity from observed wind speed and perturbed potential temperature. Gradually varying eddy thermal conductivity is generally derived from an empirical parameterization scheme. We utilize wind speed and potential temperature measurements, along with the variational assimilation technique, to de- rive this parameter. The objective function is constructed by the square of the differences between the observation and model value. The new method is validated by numerical experiments with simulated measurements, revealing that the order of the root mean squre error is 10-2 and thus confirming the method＇s robustness. In addition, this me- thod is caoable of anti-interference, as it effectivelv reduces the influence of observation error.
基金supported by the State Key Laboratory of Advanced Optical Communication Systems and Networks (Grant No. 2008SH05)
文摘Since novel optoelectronic devices based on the peculiar behaviors of the tunneling probability, e.g., resonant tunneling devices (RTD) and band-pass filter, are steadily proposed, the analytic transfer matrix (ATM) method is extended to study these devices. For several examples, we explore the effect of the scattered subwaves on tunneling; it is shown that the resonant or band-pass structures in tunneling probability are determined by the phase shift results from the scattered subwaves.
文摘In this paper,by the WKB method the relation between the energy increase of internal inertial gravity waves and heterogeneous atmospheric stratification is derived,and a new generalized wave action is defined and its conservation is proved.
基金This research is supported by the National Natural Science Foundation of China.
文摘In this paper the influences of nonuniform stratification on the propagating paths of internal inertial-gravity and pure gravity wave energy are discussed by using the WKB approximation method.The conditions for conservation of wave energy,generalized wave action and wave enstrophy are obtained.The necessary condition of instability for inter- nal gravity waves and the equation governing the refraction of wave rays are derived.Two types of critical levels are giv- en.Finally,the wave rays for different distributions of stratification are calculated by using the fourth-order Runge-Kutta method.
