Where Is Phase Velocity in Minkowski Space?

In the special theory of relativity, massive particles can travel at neither the speed of light c nor faster. Meanwhile, since the photon was quantized, many have thought of it as a point particle. How pointed? The idea could be a mathematical device or physical simplification. By contrast, the preceding notion of wave-group duality has two velocities: a group velocity vg and a phase velocity vp. In light vp = vg = c;but it follows from special relativity that, in massive particles, vp > c. The phase velocity is the product of the two best measured variables, and so their product constitutes internal motion that travels, verifiably, faster than light. How does vp then appear in Minkowski space? For light, the spatio-temporal Lorentz invariant metric is s2=c2t2−x2−y2−z2, the same in whatever frame it is viewed. The space is divided into 3 parts: firstly a cone, symmetric about the vertical axis ct > 0 that represents the world line of a stationary particle while the conical surface at s = 0 represents the locus for light rays that travel at the speed of light c. Since no real thing travels faster than the speed of light c, the surface is also a horizon for what can be seen by an observer starting from the origin at time t = 0. Secondly, an inverted cone represents, equivalently, time past. Thirdly, outside the cones, inaccessible space. The phase velocity vp, group velocity vg and speed of light are all equal in free space, vp = vg = c, constant. By contrast, for particles, where causality is due to particle interactions having rest mass mo > 0, we have to employ the Klein-Gordon equation with s2=c2t2−x2−y2−z2+mo2c2. Now special relativity requires a complication: vp.vg = c2 where vg c and therefore vp > c. In the volume outside the cones, causality due to light interactions cannot extend beyond the cones. However, since vp > c and even vp >> c when wavelength λ is long, extreme phase velocities are then limited in their causal effects by the particle uncertainty σ, i.e. to vgt ± σ/ω, where ω is the particle angular frequency. This is the first time the phase range has been described for a massive particle.

De Broglie Wave的相速度和群速度

De Broglie Wave是可以按照几率解释的,在几率解释的基础上研究De Broglie Wave的相速度、群速度等速度特性,并建立了它们的关系。

De Broglie波的相速与群速

ＤｅＢｒｏｇｌｉｅ波的相速和群速是两类不同的速率，相速等于单色波传播速率，群速等于粒子运动速率．文中给出了相速与群速的一般关系．

Review on second-harmonic generation of ultrasonic guided waves in solid media(Ⅰ): Theoretical analyses

Considering the high sensitivity of the nonlinear ultrasonic measurement technique and great advantages of the guided wave testing method, the use of nonlinear ultrasonic guided waves provides a promising means for evaluating and characterizing the hidden and/or inaccessible damage/degradation in solid media. Increasing attention on the development of the testing method based on nonlinear ultrasonic guided waves is largely attributed to the theoretical advances of nonlinear guided waves propagation in solid media. One of the typical acoustic nonlinear responses is the generation of second harmonics that can be used to effectively evaluate damage/degradation in materials/structures. In this paper, the theoretical progress of second-harmonic generation（SHG） of ultrasonic guided wave propagation in solid media is reviewed. The advances and developments of theoretical investigations on the effect of SHG of ultrasonic guided wave propagation in different structures are addressed. Some obscure understandings and the ideas in dispute are also discussed.

Relating the probability distribution of a de Broglie wave to its phase velocity

We show that the phase velocity in a stationary state of a de Broglie wave can be directly obtained from the probability distribution, i.e. the quantum trajectories, without detailed knowledge of the phase term itself. In other words, the amplitude of a de Broglie wave function describes not only the probability distribution but also the phase velocity distribution. Using this relationship, we comment on two calculations of the Goos-H nchen shift in de Broglie waves.

A Travelling Wave Group II: Antiparticles in a Force Field

The travelling wave group is a solution to the wave equation. With a Gaussian envelope, this stable wave does not spread as it propagates. The group is derived for electromagnetic waves and converted with Planck’s law to quantized photons. The resulting wave is a probability amplitude, and this is adapted to particles subject to special relativity. By including mass and by inverting the wave group, a description for antiparticles is derived. The consequent explanation is consistent with Dirac’s relativistic equation and with his theory of the electron;while being more specific than his idea of the wave packet, and more stable. The travelling wave group is extended to describe the positron, either free or in an external field.

A Wave Group for Entanglement, Linking Uncertainties in Time and Space

A carrier wave in a 5-dimensional wave group is examined for information on electromagnetic waves and on particle probability amplitudes. Simulations by Maxwell’s equations show that the phase and group velocities in electromagnetic waves are equal, both in vacuo and in dielectric media. By contrast, particle probability amplitudes in wave packet motion are more complicated. A dependence of rest mass on relative phase and group velocities is derived by consistency. Occurrences that are simultaneous and connected on wave fronts in the rest frame, appear separated when observed in moving frames. Uncertainties in space and time are linked by the probability amplitude wave group.

A Travelling Wave Group III—Consistent with QED

The travelling wave group is a stable wave packet. Many surprising results are derived from it. The group is easily quantized for photons and applied, as a solution to the relativistic Klein-Gordon equation, to free particles. Further solutions to the resulting algebraic equation provide a stable wave function for free antiparticles. Consistency with the superstructure of quantum electrodynamics is obtained by an assignment to the electron antiparticle of negative mass and negative charge. Then in 5-dimensional space-time-mass, CPT invariance transforms to M’PT conservation in either charged or neutral particles, while many other consequences are also evident.

On the Existence of a Minimum Universal Speed of Physical Transmissions Associated with Matter Wave in Special Relativity

In this work, we show that it is possible to establish coordinate transformations between inertial reference frames in the theory of special relativity with a minimum universal speed of physical transmissions. The established coordinate transformations, referred to as modified Lorentz transformations because they have almost identical form to the Lorentz transformations, also comply with the requirement of invariance of the Minkowski line element. Particularly, the minimum universal speed can be associated with the phase speed of de Broglie matter wave. As application, we also discuss the possibility to formulate relativistic classical and quantum mechanics for the special relativity associated with the modified Lorentz transformations, which describes physical processes that represent an expansion or a collapsing of massive quantum particles.

新型单相超材料的低频带隙和振动特性研究

低频振动和噪声的抑制一直是工程中的重要问题。本文提出了一种新型的单相超材料结构,有望应用于噪声控制。基于布洛赫(Bloch)定理和有限元方法(FEM),数值分析了这种新型结构的带隙特性和带隙形成机制,参数化分析了带隙与几何尺寸的依赖关系,用三维色散曲面、频率等值线、相速度和群速度等探究了波在结构中的传播特性,而且分析了有限尺寸晶格中的弹性波衰减,验证了带隙的存在对弹性波的有效抑制作用。结果表明,该新型结构具备优异的带隙特性,带隙对几何尺寸较敏感,可由此实现带隙的可调性,该新型结构能对1000Hz频率以内的弹性波起到很好的抑制。为低频超材料结构的设计开辟了一条思路。

Study on characteristics of internal solitary waves in the Malacca Strait based on Sentinel-1 and GF-3 satellite SAR data

The study of the characteristics of internal solitary waves happened in the Malacca Strait is an urgent problem for submarine, ship navigation and marine engineering. Based on SAR remote sensing data obtained from the high spatial resolution Sentinel-1 and GF-3, the internal solitary wave characteristics of the Malacca Strait are investigated. By use of 20 Sentinel-1 SAR images from June 2015 to December 2016 and 24 GF-3 images from April2018 to March 2019, the spatial distribution characteristics of internal solitary wave s are statistically analyzed. It is found that the internal solitary waves are usually in the form of wave packets and single solitary waves, and the maximum crest length of the leading wave can reach 39 km. The amplitude and group velocity of internal solitary wave s can be calculated by the inversion model of high-order nonlinear Schrodinger(NLS) equation, and the calculated amplitude of the internal solitary wave s and the propagation group velocity of the wave packets range from 4.7 m to 23.9 m and 0.12 m/s to 0.40 m/s, respectively. The range of phase velocity of single internal solitary waves obtained by Kd V equation is 0.26 m/s to 0.60 m/s. In general, the amplitude and the velocity of internal solitary wave s in Malacca strait are related to the topography.

Time-domain analysis of second-harmonic generation of primary Lamb wave propagation in an elastic plate

Within the second-order perturbation approximation, this paper investigates the physical process of generation of the time-domain second harmonic by a primary Lamb wave waveform in an elastic plate. The present work is performed based on the preconditions that the phase velocity matching is satisfied and that the transfer of energy from the primary Lamb wave to the double frequency Lamb wave is not zero. It investigates the influences of the difference between the group velocities of the primary Lamb wave and the double frequency Lamb wave, the propagation distance and the duration of the primary Lamb wave waveform on the envelope shape of the time-domain second harmonic. It finds that the maximum magnitude of the envelope of the second-harmonic waveform can grow within some propagation distance even if the condition of group velocity matching is not satisfied. Our analyses also indicate that the maximum magnitude of the envelope of the second-harmonic waveform is kept constant beyond a specific propagation distance. Furthermore, it concludes that the integration amplitude of the time-domain second-harmonic waveform always grows with propagation distance within the second-order perturbation. The present research yields new physical insight not previously available into the effect of generation of the time-domain second harmonic by propagation of a primary Lamb wave waveform.

Quantum Mechanics: Internal Motion in Theory and Experiment

Dispersion dynamics applies wave-particle duality, together with Maxwell’s electromagnetism, and with quantization E = hν = ħω (symbol definitions in footnote) and p = h/λ = ħk, to special relativity E2 = p2c2 + m2c4. Calculations on a wave-packet, that is symmetric about the normal distribution, are partly conservative and partly responsive. The complex electron wave function is chiefly modelled on the real wave function of an electromagnetic photon;while the former concept of a “point particle” is downgraded to mathematical abstraction. The computations yield conclusions for phase and group velocities, vp⋅vg = c2 with vp ≥ c because vg ≤ c, as in relativity. The condition on the phase velocity is most noticeable when p≪mc. Further consequences in dispersion dynamics are: derivations for ν and λ that are consistently established by one hundred years of experience in electron microscopy and particle accelerators. Values for vp = νλ = ω/k are therefore systematically verified by the products of known multiplicands or divisions by known divisors, even if vp is not independently measured. These consequences are significant in reduction of the wave-packet by resonant response during interactions between photons and electrons, for example, or between particles and particles. Thus the logic of mathematical quantum mechanics is distinguished from experiential physics that is continuous in time, and consistent with uncertainty principles. [Footnote: symbol E = energy;h = Planck’s constant;ν = frequency;ω = angular momentum;p = momentum;λ = wavelength;k = wave vector;c = speed of light;m = particle rest mass;vp = phase velocity;vg = group velocity].

Previously Unknown Formulas for the Relativistic Kinetic Energy of an Electron in a Hydrogen Atom

Einstein’s energy-momentum relationship, which holds in an isolated system in free space, contains two formulas for relativistic kinetic energy. Einstein’s relationship is not applicable in a hydrogen atom, where potential energy is present. However, a relationship similar to that can be derived. That derived relationship also contains two formulas, for the relativistic kinetic energy of an electron in a hydrogen atom. Furthermore, it is possible to derive a third formula for the relativistic kinetic energy of an electron from that relationship. Next, the paper looks at the fact that the electron has a wave nature. Five more formulas can be derived based on considerations relating to the phase velocity and group velocity of the electron. This paper presents eight formulas for the relativistic kinetic energy of an electron in a hydrogen atom.

量子力学中的2种波包

提出在量子力学中存在2种不同层面上的波包定义.从德布罗意波长的推导入手分析了第1层面上的波包合成,阐述了其在薛定谔方程推导过程中的应用,指出2种情形下对能量的不同定义所导致的相速度取值的巨大差异.利用一维自由电子运动的波包模型介绍了基于态叠加原理的第2层面上的波包合成,指出此层面上波包合成中子波的态速度就是第1层面波包合成中的群速度.量子力学中微粒的描述需要2个层面上波包概念的同时运用,由于2个层面上波包的相速度与群速度、态速度之间都不相等,说明了量子力学波包是不稳定的.该研究对量子力学的学习者理解波态叠加原理、波粒二象性以及量子力学的概率复数表达之间的内在联系有指导意义.

中国东北地区噪声层析成像

中国东北地区是中国唯一的深震孕育区，获取该区的壳幔结构，对于研究板块俯冲、深震以及火山活动等有重要的科学意义.本文利用该区159个固定台站2011年1月至2012年6月和27个流动台站2011年1月至2011年6月间的垂向波形连续记录，计算了台站间的预估格林函数，并采用基于连续小波变换的时频分析方法，测量了双台路径上瑞雷波的群速度和相速度频散曲线.通过质量控制和筛选，最终得到了2204条路径上周期5~40 s的群/相速度频散曲线.检测板测试表明，反演结果可以达到2°×2°的分辨.利用Ditmar ＆ Yanovskaya反演方法，我们得到了研究区（105°E—135°E，39°N—52°N）周期8~30 s的瑞雷波的群速度和相速度分布图.不同周期的群/相速度分布图，反映了不同深度S波速度的横向变化情况.研究结果显示：中国东北地区的地壳上地幔S波速度结构存在横向非均匀性.短周期（如8 s）的群/相速度分布与地表地质构造具有明显的相关性，具体来说，山区显示为高速，沉积盆地显示为低速；随着周期的增大（如15 s，22 s），地形的控制作用相对减弱；较长周期（如30 s）的群/相速度分布与地壳厚度密切相关.

中国东北地区地壳上地幔三维S波速度结构

收集了中国东北地区159个固定地震台2011年1月至2012年6月和27个流动地震台2011年1月至2011年6月间的垂向连续记录,根据噪声成像方法得到研究区（105°E—135°E,39°N—52°N）较短周期（8∽30 s）的瑞雷波群速度和相速度频散资料,再结合该区已有的天然地震长周期瑞雷波（36∽145 s）的群速度频散资料,我们反演得到了中国东北地区200 km以浅深度范围内的三维壳幔S波速度结构,并得到了该区的岩石圈厚度分布图.结果表明：研究区中、下地壳S波速度结构的横向分布,在重力梯度带两侧有很大的不同,以东地区显示为大范围的高速,以西地区则呈现为大面积的低速;松辽盆地下方岩石圈地幔表现为显著的高速,岩石圈地幔底界面深度可能在90∽100 km,薄的岩石圈盖层暗示东北地区的岩石圈可能发生了减薄;郯庐大断裂下方呈现出大范围的比较显著的低速特征,断裂下方上地幔顶部可能有热物质活动.

半解析有限元法求解钢轨中超声导波频散曲线

钢轨中超声导波的频散曲线是采用超声导波技术进行无缝线路钢轨完整性检测的重要参考依据。通过传统的有限元模态分析方法,无法求解得到钢轨中超声导波完整的频散曲线,针对这一问题,采用了半解析有限元方法。求解时假设导波在钢轨中以简谐振动的方式传播,仅对钢轨的横截面采用三角形单元进行有限元网格划分,经理论推导得到超声导波在钢轨中传播的波动方程,通过求解特征方程,得到波数与频率的值,进而获取频率与相速度、群速度的关系,绘制出频散曲线。通过求解得到的特征向量还可以分析各导波模态的振动特性。实验结果表明,半解析有限元法求解得到的我国无缝线路CHN60钢轨的频散曲线与实际线路测试结果有很好的一致性。

半解析有限元法分析兰姆波频散特性

为了获得各向同性均质自由铝板中兰姆波的频散特性曲线,采用了半解析有限元方法。此方法仅需要在自由板的横截面进行有限元离散,而假定兰姆波沿波的传播方向以简谐振动的方式运动,由哈密顿原理推导出兰姆波在自由板中传播的波动方程,通过特征方程求解得到波数和频率的关系。通过半解析有限元方法可以绘制各向同性均质自由铝板中兰姆波的相速度频散曲线,不需模态跟踪,由特征值和特征向量可直接求解得到群速度频散曲线,由特征向量可进行兰姆波的振动模态分析,这种方法还可推广用于任意横截面形状波导介质中导波模态的求解。

近地表弹性介质瑞雷波勘探研究进展与展望

瑞雷波法已在研究地球内部结构、近地表地球物理工程和超声无损检测等领域中获得了广泛应用,尤其是近年来瑞雷波法作为近地表场地表征新兴领域的前沿技术已成为国际学术研究与应用的热点。对国内外近地表弹性介质瑞雷波勘探的主要研究成果与进展进行了综述,通过对瑞雷波勘探现有的研究成果和进展密切追踪发现当前近地表瑞雷波勘探主要基于水平地表弹性水平层状介质模型,利用单分量瑞雷波相速度频散曲线单目标反演获得一维横波速度剖面和相关岩土力学参数。但是,该方法也存在着现有瑞雷波频散曲线反演极易出现模式误识别、现有单分量单目标瑞雷波反演未充分利用多分量信息、现有瑞雷波相速度反演未充分利用群速度传播特性、现有瑞雷波反演未充分利用质点椭圆极化振动特性等挑战性学术难题和不足。基于上述问题,建议未来近地表弹性介质瑞雷波勘探重点研究方向应集中在进行多模式表面波全速度谱反演研究、多站多分量表面波相速度多目标全速度谱反演研究、单站多分量表面波群速度多目标全速度谱反演研究和单站多分量表面波椭圆极化振动特性多目标反演研究。由此构建新的近地表多分量瑞雷波多目标全速度谱反演理论,引领多分量瑞雷波多目标反演学科前沿,拓展现有单分量瑞雷波单目标反演理论范畴,推动多分量瑞雷波高精度实用勘探技术的发展。