WavewatchⅢ模拟和统计方法在最大波高预报方面的评测分析

为了研究WavewatchⅢ(WWⅢ)海浪模型对最大波高的模拟能力及其与传统统计关系方法的差异,通过对两次台风浪过程的后报模拟和半年的业务化预报,分析了WWⅢ数值模拟的准确度及其与统计关系方法的精度差异。研究结果表明:WWⅢ数值模拟的最大波高(Hmax)的精度略低于有效波高(Hs),但也达到了24 h预报相对误差(H_(max)≥1 m)低于18%、相关系数高于0.94的水平,模拟精度可靠,可以用于业务化预报;与两种统计关系方法(H_(max)和H_(s)分别为1.42和1.52)计算的最大波高相比,数值模拟的精度总体与其相当,但在H_(max)和H_(s)比值大于1.65这种易出现危险的海况下,数值模拟具有更高的准确性,更适合应用于海浪预警报服务。

基于D-Wave Advantage的量子退火公钥密码攻击算法研究

D-Wave专用量子计算机的原理量子退火凭借独特的量子隧穿效应可跳出传统智能算法极易陷入的局部极值,可视为一类具有全局寻优能力的人工智能算法.本文研究了两类基于量子退火的RSA公钥密码攻击算法(分解大整数N=pq):一是将密码攻击数学方法转为组合优化问题或指数级空间搜索问题,通过Ising模型或QUBO模型求解,提出了乘法表的高位优化模型,建立新的降维公式,使用D-Wave Advantage分解了 200万整数2269753.大幅度超过普渡大学、Lockheed Martin和富士通等实验指标,且Ising模型系数h范围缩小了 84%,系数J范围缩小了 80%,极大地提高了分解成功率,这是一类完全基于D-Wave量子计算机的攻击算法;二是基于量子退火算法融合密码攻击数学方法优化密码部件的攻击,采用量子退火优化CVP问题求解,通过量子隧穿效应获得比Babai算法更近的向量,提高了 CVP问题中光滑对的搜索效率,在D-Wave Advantage上实现首次50比特RSA整数分解.实验表明,在通用量子计算机器件进展缓慢情况下,D-Wave表现出更好的现实攻击能力,且量子退火不存在NISQ量子计算机VQA算法的致命缺陷贫瘠高原问题:算法会无法收敛且无法扩展到大规模攻击.

Nitrogen‑Doped Magnetic‑Dielectric‑Carbon Aerogel for High‑Efficiency Electromagnetic Wave Absorption

Carbonbased aerogels derived from biomass chitosan are encountering a flourishing moment in electromagnetic protection on account of lightweight,controllable fabrication and versatility.Nevertheless,developing a facile construction method of component design with carbon-based aerogels for high-efficiency electromagnetic wave absorption(EWA)materials with a broad effective absorption bandwidth(EAB)and strong absorption yet hits some snags.Herein,the nitrogen-doped magnetic-dielectric-carbon aerogel was obtained via ice template method followed by carbonization treatment,homogeneous and abundant nickel(Ni)and manganese oxide(MnO)particles in situ grew on the carbon aerogels.Thanks to the optimization of impedance matching of dielectric/magnetic components to carbon aerogels,the nitrogen-doped magnetic-dielectric-carbon aerogel(Ni/MnO-CA)suggests a praiseworthy EWA performance,with an ultra-wide EAB of 7.36 GHz and a minimum reflection loss(RLmin)of−64.09 dB,while achieving a specific reflection loss of−253.32 dB mm−1.Furthermore,the aerogel reveals excellent radar stealth,infrared stealth,and thermal management capabilities.Hence,the high-performance,easy fabricated and multifunctional nickel/manganese oxide/carbon aerogels have broad application aspects for electromagnetic protection,electronic devices and aerospace.

Tracking Regulatory Mechanism of Trace Fe on Graphene Electromagnetic Wave Absorption

Polarization and conductance losses are the fundamental dielectric attenuation mechanisms for graphene-based absorbers, but it is not fully understood in revealing the loss mechanism of affect graphene itself. For the first time, the reduced graphene oxide(RGO) based absorbers are developed with regulatory absorption properties and the absorption mechanism of RGO is mainly originated from the carrier injection behavior of trace metal Fe nanosheets on graphene. Accordingly, the minimum reflection loss(RLmin) of Fe/RGO-2composite reaches-53.38 dB(2.45 mm), and the effective absorption bandwidth achieves 7.52 GHz(2.62 mm) with lower filling loading of 2 wt%. Using off-axis electron hologram testing combined with simulation calculation and carrier transport property experiments, we demonstrate here the carrier injection behavior from Fe to graphene at the interface and the induced charge accumulation and rearrangement, resulting in the increased interfacial and dipole polarization and the conductance loss. This work has confirmed that regulating the dielectric property of graphene itself by adding trace metals can not only ensure good impedance matching, but also fully exploit the dielectric loss ability of graphene at low filler content,which opens up an efficient way for designing lightweight absorbers and may be extended to other types materials.

Efficient Electromagnetic Wave Absorption and Thermal Infrared Stealth in PVTMS@MWCNT Nano‑Aerogel via Abundant Nano‑Sized Cavities and Attenuation Interfaces

Pre-polymerized vinyl trimethoxy silane(PVTMS)@MWCNT nano-aerogel system was constructed via radical polymerization,sol-gel transition and supercritical CO_(2)drying.The fabricated organic-inorganic hybrid PVTMS@MWCNT aerogel structure shows nano-pore size(30-40 nm),high specific surface area(559 m^(2)g^(−1)),high void fraction(91.7%)and enhanced mechanical property:(1)the nano-pore size is beneficial for efficiently blocking thermal conduction and thermal convection via Knudsen effect(beneficial for infrared(IR)stealth);(2)the heterogeneous interface was beneficial for IR reflection(beneficial for IR stealth)and MWCNT polarization loss(beneficial for electromagnetic wave(EMW)attenuation);(3)the high void fraction was beneficial for enhancing thermal insulation(beneficial for IR stealth)and EMW impedance match(beneficial for EMW attenuation).Guided by the above theoretical design strategy,PVTMS@MWCNT nano-aerogel shows superior EMW absorption property(cover all Ku-band)and thermal IR stealth property(ΔT reached 60.7℃).Followed by a facial combination of the above nano-aerogel with graphene film of high electrical conductivity,an extremely high electromagnetic interference shielding material(66.5 dB,2.06 mm thickness)with superior absorption performance of an average absorption-to-reflection(A/R)coefficient ratio of 25.4 and a low reflection bandwidth of 4.1 GHz(A/R ratio more than 10)was experimentally obtained in this work.

From VIB‑to VB‑Group Transition Metal Disulfides:Structure Engineering Modulation for Superior Electromagnetic Wave Absorption

The laminated transition metal disulfides(TMDs),which are well known as typical two-dimensional(2D)semiconductive materials,possess a unique layered structure,leading to their wide-spread applications in various fields,such as catalysis,energy storage,sensing,etc.In recent years,a lot of research work on TMDs based functional materials in the fields of electromagnetic wave absorption(EMA)has been carried out.Therefore,it is of great significance to elaborate the influence of TMDs on EMA in time to speed up the application.In this review,recent advances in the development of electromagnetic wave(EMW)absorbers based on TMDs,ranging from the VIB group to the VB group are summarized.Their compositions,microstructures,electronic properties,and synthesis methods are presented in detail.Particularly,the modulation of structure engineering from the aspects of heterostructures,defects,morphologies and phases are systematically summarized,focusing on optimizing impedance matching and increasing dielectric and magnetic losses in the EMA materials with tunable EMW absorption performance.Milestones as well as the challenges are also identified to guide the design of new TMDs based dielectric EMA materials with high performance.

Accurate simulations of pure-viscoacoustic wave propagation in tilted transversely isotropic media

Forward modeling of seismic wave propagation is crucial for the realization of reverse time migration(RTM) and full waveform inversion(FWI) in attenuating transversely isotropic media. To describe the attenuation and anisotropy properties of subsurface media, the pure-viscoacoustic anisotropic wave equations are established for wavefield simulations, because they can provide clear and stable wavefields. However, due to the use of several approximations in deriving the wave equation and the introduction of a fractional Laplacian approximation in solving the derived equation, the wavefields simulated by the previous pure-viscoacoustic tilted transversely isotropic(TTI) wave equations has low accuracy. To accurately simulate wavefields in media with velocity anisotropy and attenuation anisotropy, we first derive a new pure-viscoacoustic TTI wave equation from the exact complex-valued dispersion formula in viscoelastic vertical transversely isotropic(VTI) media. Then, we present the hybrid finite-difference and low-rank decomposition(HFDLRD) method to accurately solve our proposed pure-viscoacoustic TTI wave equation. Theoretical analysis and numerical examples suggest that our pure-viscoacoustic TTI wave equation has higher accuracy than previous pure-viscoacoustic TTI wave equations in describing q P-wave kinematic and attenuation characteristics. Additionally, the numerical experiment in a simple two-layer model shows that the HFDLRD technique outperforms the hybrid finite-difference and pseudo-spectral(HFDPS) method in terms of accuracy of wavefield modeling.

Interface Engineering of Titanium Nitride Nanotube Composites for Excellent Microwave Absorption at Elevated Temperature

Currently,the microwave absorbers usually suffer dreadful electromagnetic wave absorption(EMWA)performance damping at elevated temperature due to impedance mismatching induced by increased conduction loss.Consequently,the development of high-performance EMWA materials with good impedance matching and strong loss ability in wide temperature spectrum has emerged as a top priority.Herein,due to the high melting point,good electrical conductivity,excellent environmental stability,EM coupling effect,and abundant interfaces of titanium nitride(TiN)nanotubes,they were designed based on the controlling kinetic diffusion procedure and Ostwald ripening process.Benefiting from boosted heterogeneous interfaces between TiN nanotubes and polydimethylsiloxane(PDMS),enhanced polarization loss relaxations were created,which could not only improve the depletion efficiency of EMWA,but also contribute to the optimized impedance matching at elevated temperature.Therefore,the TiN nanotubes/PDMS composite showed excellent EMWA performances at varied temperature(298-573 K),while achieved an effective absorption bandwidth(EAB)value of 3.23 GHz and a minimum reflection loss(RLmin)value of−44.15 dB at 423 K.This study not only clarifies the relationship between dielectric loss capacity(conduction loss and polarization loss)and temperature,but also breaks new ground for EM absorbers in wide temperature spectrum based on interface engineering.

A novel method for simulating nuclear explosion with chemical explosion to form an approximate plane wave: Field test and numerical simulation

A nuclear explosion in the rock mass medium can produce strong shock waves,seismic shocks,and other destructive effects,which can cause extreme damage to the underground protection infrastructures.With the increase in nuclear explosion power,underground protection engineering enabled by explosion-proof impact theory and technology ushered in a new challenge.This paper proposes to simulate nuclear explosion tests with on-site chemical explosion tests in the form of multi-hole explosions.First,the mechanism of using multi-hole simultaneous blasting to simulate a nuclear explosion to generate approximate plane waves was analyzed.The plane pressure curve at the vault of the underground protective tunnel under the action of the multi-hole simultaneous blasting was then obtained using the impact test in the rock mass at the site.According to the peak pressure at the vault plane,it was divided into three regions:the stress superposition region,the superposition region after surface reflection,and the approximate plane stress wave zone.A numerical simulation approach was developed using PFC and FLAC to study the peak particle velocity in the surrounding rock of the underground protective cave under the action of multi-hole blasting.The time-history curves of pressure and peak pressure partition obtained by the on-site multi-hole simultaneous blasting test and numerical simulation were compared and analyzed,to verify the correctness and rationality of the formation of an approximate plane wave in the simulated nuclear explosion.This comparison and analysis also provided a theoretical foundation and some research ideas for the ensuing study on the impact of a nuclear explosion.

Enhancing electromagnetic wave absorption with core‐shell structured SiO_(2)@MXene@MoS_(2)nanospheres

Material composition and structural design are important factors influencing the electromagnetic wave(EMW)absorption performance of materials.To alleviate the impedance mismatch attributed to the high dielectric constant of Ti_(3)C_(2)T_(x)MXene,we have successfully synthesized core‐shell structured SiO_(2)@MXene@MoS_(2)nanospheres.This architecture,comprising SiO_(2)as the core,MXene as the intermediate layer,and MoS_(2)as the outer shell,is achieved through an electrostatic self‐assembly method combined with a hydrothermal process.This complex core‐shell structure not only provides a variety of loss mechanisms that effectively dissipate electromagnetic energy but also prevents self‐aggregation of MXene and MoS_(2)nanosheets.Notably,the synergistic combination of SiO_(2)and MoS_(2)with highly conductive MXene enables the suitable dielectric constant of the composites,ensuring optimal impedance matching.Therefore,the core‐shell structured SiO_(2)@MXene@MoS_(2)nanospheres exhibit excellent EMW absorption performance,featuring a remarkable minimum reflection loss(RL_(min))of−52.11 dB(2.4 mm).It is noteworthy that these nanospheres achieve an ultra‐wide effective absorption bandwidth(EAB)of 6.72 GHz.This work provides a novel approach for designing and synthesizing high‐performance EMW absorbers characterized by“wide bandwidth and strong reflection loss.”

Influence of topography on the fine structures of stratospheric gravity waves:An analysis using COSMIC-2 temperature data

We derive the potential energy of gravity waves(GWs)in the upper troposphere and stratosphere at 45°S-45°N from December 2019 to November 2022 by using temperature profiles retrieved from the Constellation Observing System for Meteorology,Ionosphere,and Climate-2(COSMIC-2)satellite.Owing to the dense sampling of COSMIC-2,in addition to the strong peaks of gravity wave potential energy(GWPE)above the Andes and Tibetan Plateau,we found weak peaks above the Rocky,Atlas,Caucasus,and Tianshan Mountains.The land-sea contrast is responsible for the longitudinal variations of the GWPE in the lower and upper stratosphere.At 40°N/S,the peaks were mainly above the topographic regions during the winter.At 20°N/S,the peaks were a slight distance away from the topographic regions and might be the combined effect of nontopographic GWs and mountain waves.Near the Equator,the peaks were mainly above the regions with the lowest sea level altitude and may have resulted from convection.Our results indicate that even above the local regions with lower sea level altitudes compared with the Andes and Tibetan Plateau,the GWPE also exhibits fine structures in geographic distributions.We found that dissipation layers above the tropopause jet provide the body force to generate secondary waves in the upper stratosphere,especially during the winter months of each hemisphere and at latitudes of greater than 20°N/S.

Ion acoustic solitary waves in an adiabatic dusty plasma:Roles of superthermal electrons,ion loss and ionization

We investigate propagation of dust ion acoustic solitary wave(DIASW)in a multicomponent dusty plasma with adiabatic ions,superthermal electrons,and stationary dust.The reductive perturbation method is employed to derive the damped Korteweg-de Vries(DKdV)equation which describes DIASW.The result reveals that the adiabaticity of ions significantly modifies the basic features of the DIASW.The ionization effect makes the solitary wave grow,while collisions reduce the growth rate and even lead to the damping.With the increases in ionization cross sectionΔσ/σ_(0),ion-to-electron density ratioδ_(ie)and superthermal electrons parameterκ,the effect of ionization on DIASW enhances.

Probability Distribution Characteristics of Strong Nonlinear Waves Under Typhoon Conditions in the Northern South China Sea

The generation and propagation mechanism of strong nonlinear waves in the South China Sea is an essential research area. In this study, the third-generation wave model WAVEWATCH Ⅲ is employed to simulate wave fields under extreme sea states. The model, integrating the ST6 source term, is validated against observed data, demonstrating its credibility. The spatial distribution of the occurrence probability of strong nonlinear waves during typhoons is shown, and the waves in the straits and the northeastern part of the South China Sea show strong nonlinear characteristics. The high-order spectral model HOS-ocean is employed to simulate the random wave surface series beneath five different platform areas. The waves during the typhoon exhibit strong nonlinear characteristics, and freak waves exist. The space-varying probability model is established to describe the short-term probability distribution of nonlinear wave series. The exceedance probability distributions of the wave surface beneath different platform areas are compared and analyzed. The results show that with an increase in the platform area, the probability of a strong nonlinear wave beneath the platform increases.

Long-Term Extreme Wave Characteristics in the Water Adjacent to China Based on ERA5 Reanalysis Data

Extreme waves have a profound impact on coastal infrastructure;thus,understanding the variation law of risky analysis and disaster prevention in coastal zones is necessary.This paper analyzed the spatiotemporal characteristics of extreme wave heights adjacent to China from 1979 to 2018 based on the ERA5 datasets.Nonstationary extreme value analysis is undertaken in eight repre-sentative points to investigate the trends in the values of 50-and 100-year wave heights.Results show that the mean value of extreme waves is the largest in the eastern part of Taiwan Island and the smallest in the Bohai Sea from 1979 to 2018.Only the extreme wave height in the northeastern part of Taiwan Island shows a significant increase trend in the study area.Nonstationary analysis shows remarkable variations in the values of 50-and 100-year significant wave heights in eight points.Considering the annual mean change,E1,E2,S1,and S2 present an increasing trend,while S3 shows a decreasing trend.Most points for the seasonal mean change demon-strate an increasing trend in spring and winter,while other points show a decreasing trend in summer and autumn.Notably,the E1 point growth rate is large in autumn,which is related to the change in typhoon intensity and the northward movement of the typhoon path.

Analysis of explosion wave interactions and rock breaking effects during dual initiation

In blasting engineering, the location and number of detonation points, to a certain degree, regulate the propagation direction ofthe explosion stress wave and blasting effect. Herein, we examine the explosion wave field and rock breaking effect in terms of shockwave collision, stress change of the blast hole wall in the collision zone, and crack propagation in the collision zone. The produced shockwave on the collision surface has an intensity surpassing the sum of the intensities of the two colliding explosion shock waves. At the collisionlocation, the kinetic energy is transformed into potential energy with a reduction in particle velocity at the wave front and the wavefront pressure increases. The expansion form of the superposed shock wave is dumbbell-shaped, the shock wave velocity in the collisionarea is greater than the radial shock wave velocity, and the average propagation angle of the explosion shock waves is approximately 60°.Accordingly, a fitted relationship between blast hole wall stress and explosion wave propagation angle in the superposition area is plotted.Under the experimental conditions, the superimposed explosion wave stress of the blast hole wall is approximately 1.73 times the singleexplosionwave incident stress. The results of the model test and numerical simulations reveal that large-scale radial fracture cracks weregenerated on the blast hole wall in the superimposed area, and the width of the crack increased. The width of the large-scale radial fracturecracks formed by a strong impact is approximately 5% of the blast hole length. According to the characteristics of blast hole wallcompression, the mean peak pressures of the strongly superimposed area are approximately 1.48 and 1.84 times those of the weakly superimposedand nonsuperimposed areas, respectively.

A Novel Method for Forecasting Freak Wave Occurrence in A Short Crested Coastal Sea

In order to forecast the distribution of crest amplitudes and the occurrence of freak waves in a short crested coastal sea,a novel transformed linear simulation method is initially proposed in this paper.A Hermite transformation model expressed as a monotonic cubic polynomial serves as the foundation for the novel simulation technique.The wave crest amplitude exceedance probabilities of two sea states-one with a directional wave spectrum based on the measured wave elevation data at the Yura coast and the other with a typical directional JONSWAP wave spectrum-have been predicted using the novel simulation method that has been proposed.The likelihood that a particular critical wave crest amplitude will be exceeded is directly correlated with the probability that freak waves will occur.It is shown that the novel simulation approach suggested can provide predictions that are more precise than those obtained from the Rayleigh crest amplitude distribution model,the Jahns and Wheeler crest amplitude distribution model,or the conventional linear simulation method.This study also demonstrated that the nonlinear simulation method is less effective than the novel simulation method in terms of efficiency.

Contrasts of bimodal tropical instability waves(TIWs)-induced wind stress perturbations in the Pacific Ocean among observations,ocean models,and coupled climate models

The coupling between wind stress perturbations and sea surface temperature(SST)perturbations induced by tropical instability waves(TIWs)in the Pacific Ocean has been revealed previously and proven crucial to both the atmosphere and ocean.However,an overlooked fact by previous studies is that the loosely defined“TIWs”actually consist of two modes,including the Yanai wave-based TIW on the equator(hereafter eTIW)and the Rossby wave-based TIW off the equator(hereafter vTIW).Hence,the individual feedbacks of the wind stress to the bimodal TIWs remain unexplored.In this study,individual coupling relationships are established for both eTIW and v TIW,including the relationship between the TIW-induced SST perturbations and two components of wind stress perturbations,and the relationship between the TIW-induced wind stress perturbation divergence(curl)and the downwind(crosswind)TIW-induced SST gradients.Results show that,due to different distributions of eTIW and vTIW,the coupling strength induced by the eTIW is stronger on the equator,and that by the vTIW is stronger off the equator.The results of any of eTIW and vTIW are higher than those of the loosely defined TIWs.We further investigated how well the coupling relationships remained in several widely recognized oceanic general circulation models and fully coupled climate models.However,the coupling relationships cannot be well represented in most numerical models.Finally,we confirmed that higher resolution usually corresponds to more accurate simulation.Therefore,the coupling models established in this study are complementary to previous research and can be used to refine the oceanic and coupled climate models.

Electroacupuncture with different waveforms for primary dysmenorrhea:A randomized controlled trial

Objective To observe and compare the clinical effects of different electroacupuncture waveforms on primary dysmenorrhea.Methods This was a prospective,randomized,three-group,parallel-controlled trial.Participants with primary dysmenorrhea were randomly divided into dense-sparse wave,continuous wave,and discontinuous wave groups in a 1:1:1 ratio.Two lateral Ciliao(BL 32)points were used.All three groups started treatment 3–5 days before menstruation,once a day for six sessions per course of treatment,one course of treatment per menstrual cycle,and three menstrual cycles.The primary outcome measure was the proportion with an average visual analog scale(VAS)score reduction of≥50%from baseline for dysmenorrhea in the third menstrual cycle during treatment.The secondary outcome measures included changes in dysmenorrhea VAS scores,Cox Menstrual Symptom Scale scores and the proportion of patients taking analgesic drugs.Results The proportion of cases where the average VAS score for dysmenorrhea decreased by≥50%from baseline in the third menstrual cycle was not statistically significant(P>.05).Precisely 30 min after acupuncture and regarding immediate analgesia on the most severe day of dysmenorrhea,there was a statistically significant difference in the dense-sparse wave group compared with the other two groups during the third menstrual cycle(P<.05).Additionally,there was a statistically significant difference between the dense-sparse wave and discontinuous wave groups 24 h after acupuncture(P<.05).Conclusions Waveform electroacupuncture can alleviate primary dysmenorrhea and its related symptoms in patients.The three groups showed similar results in terms of short-and long-term analgesic efficacy and a reduction in the number of patients taking analgesic drugs.Regarding achieving immediate analgesia,the dense-sparse wave group was slightly better than the other two groups.

Blast waveform tailoring using controlled venting in blast simulators and shock tubes

A critical challenge of any blast simulation facility is in producing the widest possible pressure-impulse range for matching against equivalent high-explosive events.Shock tubes and blast simulators are often constrained with the lack of effective ways to control blast wave profiles and as a result have a limited performance range.Some wave shaping techniques employed in some facilities are reviewed but often necessitate extensive geometric modifications,inadvertently cause flow anomalies,and/or are only applicable under very specific configurations.This paper investigates controlled venting as an expedient way for waveforms to be tuned without requiring extensive modifications to the driver or existing geometry and could be widely applied by existing and future blast simulation and shock tube facilities.The use of controlled venting is demonstrated experimentally using the Advanced Blast Simulator(shock tube)at the Australian National Facility of Physical Blast Simulation and via numerical flow simulations with Computational Fluid Dynamics.Controlled venting is determined as an effective method for mitigating the impact of re-reflected waves within the blast simulator.This control method also allows for the adjustment of parameters such as tuning the peak overpressure,the positive phase duration,and modifying the magnitude of the negative phase and the secondary shock of the blast waves.This paper is concluded with an illustration of the potential expanded performance range of the Australian blast simulation facility when controlled venting for blast waveform tailoring as presented in this paper is applied.

Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation

For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.