On modelling an iceberg embedded in shore-fast sea ice

The effect of long ice-coupled waves impinging on a tabular iceberg, an ice island or a thick sea ice floe trapped within a thin veneer of shore fast sea ice of substantial extent is considered. The waves most likely originate as ocean waves in the open sea beyond the fast ice boundary, from where they propagate into the sea ice. There their character is altered because of the flexural properties of the ice. The geophysical/engineering problem posed is solved by a Green's function method that redevelops, for a different surface boundary condition, an earlier study concerned with a freely floating ice floe. Reflection and transmission coefficients for the berg are found to depend strongly on its thickness and length. Amongst other things, the work relates to the operational safety of natural and artificially thickened Arctic ice platforms located in a contiguous ice sheet.     

Flexural gravity wave over a floating ice sheet near a vertical wall

The behavior of flexural gravity waves propagating over a semi-infinite floating ice sheet is studied under the assumptions of small amplitude linear wave theory. The vertical wall is assumed to be either fixed or harmonically oscillating with constant horizontal displacement, in which case the problem is analogous with a harmonically oscillating plane vertical wavemaker. The potential flow approach is adhered to and the higher-order mode–coupling relations are applied to determine the unknown coefficients present in the Fourier expansion formula of the potential functions. The ice sheet is modeled as a thin semi-infinite elastic beam. Three different edge conditions are applied at the finite edge of the floating ice sheet. The effects of different edge conditions, the thickness of the ice sheet and the water depth on the surface strain, the shear force along the ice sheet, the horizontal force on the vertical wall, and the flexural gravity wave profile are analyzed in detail.     

Dynamic strain response of lake and sea ice to moving loads

The results from two experiments to measure the strains due to a vehicle moving over ice are discussed in the context of theoretical work derived from existing solutions in the literature. The experiments took place on two very different types of ice; the lake ice of Femund in Norway, and sea ice near Scott Base in the Antarctic. In both cases, strain was measured directly by means of strainmeters developed specifically for use on ice. The existence of a critical velocity at which the strain is resonant is discussed, and using values derived from the data, a dispersion equation for free waves is solved in the super-critical domain to provide wavelength estimates. At subcritical speeds a moving static load calculation provides the equivalent theory. The experimental results for lake ice and sea ice are similar, although some differences do exist. The magnification factor of the critical strain over static strain is, for example, larger for lake ice (2.25) than for sea ice (1.45). A critical velocity of 15.2 ms^?1 was observed for lake ice, for sea ice the value was 19.6 ms^?1; both are associated with the minimum phase velocity of free wave propagation.     

Azimuth dependence of Doppler spectra of sea clutter at low grazing angle

The effect of varying the azimuth angle of a cliff-top radar on the Doppler spectra of low grazing angle sea clutter has been investigated. The peak of the horizontally polarised Doppler spectra was found to depend on the component of the phase velocity of the breaking waves in the line of sight of the radar. The analysis was presented in terms of two main scattering components, one slow and one fast, which could explain the Doppler spectra as well as the higher moments of the spectra. The connection to the amplitude statistics for the two components was also discussed. The slow component was present all the time and had amplitude statistics that were Rayleigh-distributed. The fast component was present only intermittently and had amplitude statistics that were spikier than the Rayleigh distribution. The width of the spectra varied with azimuth in a way unrelated to the direction relative to the wind. A wave tank experiment with long wavelength mechanical waves and wind-generated waves illustrated that the cause of the broadening of the spectra was probably due to the effect of swell waves.     

Deflection of a floating sea ice sheet induced by a moving load

A large number of time-deflection curves was obtained from a field experiment carried out on the sea ice sheet at Lake Saroma, Hokkaido, Japan, using a Skidoo snowmobile as a moving load with the test speed changed from 0 to 14.2 m/s. The thickness of the ice sheet was 0.17–0.18 m, the ice temperature was ?2.5 to ?4°C at the surface, and the water depth was 6.8 m beneath the ice sheet. The experimental results indicated the existence of a critical speed u_c and its value was estimated to be 5.8 m/s. At speed above it two ice waves were generated, one ahead of and the other behind the vehicle. A variation of wavelength with vehicle speed was expressed by a modified dispersion relation for the flexural ice wave in a floating ice sheet. The ice deflection pattern was classified into five stages. At u_c the depression around the vehicle was maximal in depth and minimal in width. The center of the depression began to lag behind the vehicle even at fairly low speeds below u_c.     

Granular flow in the marginal ice zone

The region of sea ice near the edge of the sea ice pack is known as the marginal ice zone (MIZ), and its dynamics are complicated by ocean wave interaction with the ice cover, strong gradients in the atmosphere and ocean and variations in sea ice rheology. This paper focuses on the role of sea ice rheology in determining the dynamics of the MIZ. Here, sea ice is treated as a granular material with a composite rheology describing collisional ice floe interaction and plastic interaction. The collisional component of sea ice rheology depends upon the granular temperature, a measure of the kinetic energy of flow fluctuations. A simplified model of the MIZ is introduced consisting of the along and across momentum balance of the sea ice and the balance equation of fluctuation kinetic energy. The steady solution of these equations is found to leading order using elementary methods. This reveals a concentrated region of rapid ice flow parallel to the ice edge, which is in accordance with field observations, and previously called the ice jet. Previous explanations of the ice jet relied upon the existence of ocean currents beneath the ice cover. We show that an ice jet results as a natural consequence of the granular nature of sea ice.     

Wave-generated fractures in river ice covers

The breakup of the ice cover in northern rivers is a brief but crucial event in the life cycle of many aquatic species and can trigger extreme ice jam events with major socio-economic impacts and significant climate change implications. An important, but vaguely understood, breakup process is the fracture of the winter ice cover by low-amplitude water waves. Previous work on this subject has been based on the assumption of an infinitely long wave propagating under an infinitely long and “edgeless” ice cover. This configuration does not account for structural constraints imposed by the proximity of an ice edge or a transverse crack. Consequently, it only furnishes approximate values of bending stresses, and tells little about the spacing of cracks that may be generated by an advancing wave, which is the only visual evidence that can identify the relevant fracture mechanism in the field. Herein, edge proximity is taken into account by making plausible simplifications to the ice response equation, and using wave forms of limited extent. It is shown that such conditions generally produce higher bending stresses than does the infinite wave/edgeless cover configuration. The distance of the peak bending stress from the edge, which defines the spacing of cracks, varies with wavelength and is less than 100 ice thicknesses or so. This is comparable to that of high-amplitude, single waves (or surges) that result from ice jam releases, but much less than fractures generated by bending on horizontal planes, caused by the meandering river plan form. Comparison of the present results with the limited available evidence indicates that wave-generated fractures occur during the passage of ice jam release surges.     

Wave forces on steeply-sloping sea walls: oblique incidence

Previous work, [3], studying the forces exerted by non-breaking, normally-incident water waves of small amplitude on a sloping sea wall is here extended to the case of oblique incidence. The range of applicability of the Galerkin solutions is increased by means of the Shanks transform. Results are presented for a planar, outward-sloping sea wall. In shallow water, the total normal wave force per unit span is found to decrease as the wall slope increases, except for extremely obliquely incident waves. In deep water, it increases. Regarded as a function of the angle of incidence , the wave force in shallow water is virtually independent of , except for very oblique waves. In deep water, by contrast, the force first increases with and then decreases. In this case, the maximum wave force does not occur for normally incident waves.     

Sea ice thickness measurement using episodic infragravity waves from distant storms

The thinning and retreat of Arctic sea ice is one of the most dramatic manifestations of recent climate warming. Though ice extent can be routinely monitored by satellite, ice thickness is far more difficult to measure operationally. We show that small amplitude, long period waves — termed infragravity waves — can be used to measure ice thickness at basin scales by determining their travel time between measurement sites. The waves travel at a different speed in ice than in open water, the difference being a sensitive function of ice thickness. We present measurements from near the North Pole where the travel time of 15 s waves is reduced by around 7 h for a typical 2 m ice thickness. Our results demonstrate that a basin-scale observation network which can track the effect of global change on Arctic sea ice thickness is practical and feasible using current technology.     

Reflection of plane waves at the boundary of a pre-stressed elastic half-space subject to biaxial stretch and simple shear

This paper is concerned with the effect of finite pure homogeneous biaxial stretch together with simple shear deformation on the reflection from a plane boundary of elastic waves propagating in a half-space of incompressible isotropic elastic material. This generalizes the previous work in which, separately, pure homogeneous strain and simple shear were considered. For a special class of constitutive law, it is shown that an incident plane harmonic wave propagating in the considered plane gives rise to a surface wave in addition to a reflected wave for every angle of incidence. The amplitude of the surface wave may vanish at certain discrete angles depending on the state of stress, biaxial stretch and simple shear deformation and then specialized to recover results obtained previously. The amplitude of the reflected wave is independent of the pre-stress but does depend upon the magnitude of deformation under consideration. The dependence of the reflected and surface wave behavior on the angle of incidence, amount of shear strain, biaxial stretch and the state of stress is illustrated graphically.     

Reflection and transmission of waves from a plane interface between two microstretch solid half-spaces

In this paper, we have investigated the wave propagation and their reflection and transmission from a plane interface between two different microstretch elastic solid half-spaces in perfect contact. It is shown that there exist five waves in a linear homogeneous isotropic microstretch elastic solid, one of them travel independently, while other waves are two sets of two coupled waves. It is also shown that these waves travel with different velocities, three of which disappear below a critical frequency. Amplitude ratios and energy ratios of various reflected and transmitted waves are presented when a set of coupled longitudinal waves and a set of coupled transverse waves is made incident. It is found that the amplitude ratios of reflected and transmitted waves are functions of angle of incidence, frequency and are affected by the elastic properties of the media. Some special cases have been reduced from the present formulation.     

Reflection of plane waves at the free surface of a fibre-reinforced elastic half-space

The propagation of plane waves in fibre-reinforced, anisotropic, elastic media is discussed. The expressions for the phase velocity of quasi-P (qP) and quasi-SV (qSV) waves propagating in a plane containing the reinforcement direction are obtained as functions of the angle between the propagation and reinforcement directions. Closed form expressions for the amplitude ratios for qP and qSV waves reflected at the free surface of a fibre-reinforced, anisotropic, homogeneous, elastic half-space are obtained. These expressions are used to study the variation of amplitude ratios with angle of incidence. It is found that reinforcement has a significant effect on the amplitude ratios and critical angle     

近岸波浪破碎区污染物运动的数值模拟

波浪斜向近岸传播中由波浪破碎作用所引起的近岸波流体系是近岸非常重要的环境水动力因素,对该区域污染物的运动分布起着重要的影响作用。结合近岸波浪传播的抛物型缓坡方程、近岸波流场数学模型及污染物在近岸波流作用下运动的数学模型,对近岸斜向入射波浪破碎所产生的沿岸波流场中污染物的分布运动过程进行了数值模拟,并结合实验结果对数值模拟结果进行了分析验证。     

Wave propagation passing over a submerged porous breakwater

A linear model of waves propagating over a submerged porous breakwater is derived from two coupled boundary-value problems, each of which represents the governing equation in a different medium. The model is similar to the shallow-water equations (SWE), with a damping term proportional to the character of the porous breakwater. Therefore, waves traveling above the breakwater will be absorbed, and the amplitude decreases. The wave propagation passing over the submerged breakwater for monochromatic and solitary waves is analyzed. For monochromatic waves, the numerical solution agrees with the analytical. The amplitude decreases exponentially with respect to the space variable in the region above the breakwater. The reflected wave is also analyzed when the model is combined with a model using the shallow-water equations.     

Experimental study on surface wave propagating through a grease–pancake ice mixture

A laboratory experiment was carried out in two wave tanks in a cold room. This experiment was designed to better understand the dispersion relation and amplitude attenuation for surface waves propagating through ice covers consisting of a grease and pancake ice mixture. Each test started from a calm condition to avoid reflection effects. The portion of the wave signals before the arrival of reflections was used to measure the wave number and the attenuation rate. The comparison between experimental results and the two-layer viscous model (Keller, 1998) indicates that modeling a grease–pancake ice cover as a viscous fluid is not sufficient to describe the observed dispersion relation and amplitude attenuation. Extended models including other mechanisms may be necessary to better describe wave–ice interactions for a grease–pancake mixture.     

Investigation of the kinetics of cracks in polymethyl methacrylate by dynamic photoelasticity

Methods have been developed for experimental investigation of the kinetics of fast cracks in plates of optically active model material making it possible to load by pulsed pressure varying the amplitude and length of the pulse and also to use the photoelastic method for recording dynamic processes with use of a superfast motion picture camera. The relationship of the dynamic stress intensity factor to crack growth rate and plate thickness was obtained taking into consideration the influence of longitudinal and transverse elastic waves. A clear relationship between stress intensity factor and v_cr was established. It was shown that with a fast crack growth rate (v_cr < 270 m/sec) the uniqueness between stress intensity factor and v_cr is disturbed by the action of reflected waves. The character of action on the stress intensity factor and crack growth rate of longitudinal and transverse waves reflected from the boundaries of the plate and also the influence of inertial effects caused by stress waves, high crack growth rates, and a change in the stressed and strained state with an increase in plate thickness were determined. It was determined that in polymethyl methacrylate in incidence on a crack at an angle of 90° a longitudinal wave increases the stress intensity factor and v_cr (with v_cr<625 m/sec) while a transverse wave decreases these characteristics.Translated from Problemy Prochnosti, No. 9, pp. 40–46, September, 1991.     

P1波在非饱和土地基表面的反射特性

非饱和土是地表天然土体中最常见的一种形式。与饱和土不同,非饱和土内可传播四种体波:一种剪切波(S波)和三种压缩波(P1波、P2波、P3波)。其中P1传播特性与饱和土中的快压缩波相似:传播速度最快,且衰减系数最低。因此,该文运用非饱和孔隙介质理论阐述了弹性波在非饱和土中的传播特性,分析了平面P1波在非饱和土介质自由边界上的反射问题。根据边界条件,分别导出了在非饱和土自由边界上的四种反射波:反射P1波、反射P2波、反射P3波及反射S波的振幅反射系数及能量反射系数的理论表达式,并且在此基础上进行了数值计算。算例中讨论了四种反射波的振幅反射系数及能量反射系数受P1波入射角及地基土体饱和度变化的影响情况。结果表明:各反射波的振幅反射系数及能量反射系数不仅与入射角有关,也受到饱和度变化的影响。这些结论对土动力学的理论研究以及工程地震勘探具有一定指导意义。     

Anomalous propagation of elastic waves through a boundary between liquid and magnetoacoustic material

We consider a longitudinal acoustic wave incident onto a plane boundary between a liquid and a magnetoacoustic medium representing an antiferromagnetic material with anisotropy of the easy plane type, occurring in the vicinity of an orientational phase transition with respect to magnetic field. The directions of propagation and the amplitudes of reflected and transmitted longitudinal and transverse waves are determined. The possibility of an effective field control for the refraction angle and the wave type transformation is demonstrated. Beginning with a certain critical angle of incidence, the longitudinal and, eventually, the transverse waves in the magnetic medium become inhomogeneous and slide along the interface. If the magnetic material is sufficiently close to the phase transition point, the waves can be reirradiated into the liquid medium.     

Diffraction of horizontally polarized ultrasonic plane waves on a periodically corrugated solid-liquid interface for normal incidence and Brewster angle incidence

The theory, and the use at normal incidence, of shear-vertically polarized waves (with polarization vector in the plane containing the incident wave vector and the normal on the interface) using the mode conversion method has been tackled by others. Here we develop the theory for shear-horizontally polarized incident waves (with polarization vector perpendicular to both the normal on the interface and the incoming wave vector). We take into account normal incidence as well as oblique incidence. For normal incidence, we discover the generation of Love waves. If oblique incidence is considered, we discover the existence of a Brewster angle of incidence, comparable with the Brewster angle in optics, in which a diffraction grating can be used as a polarization filter.     

One-dimensional plastic materials with work-hardening

Summary Acceleration waves in one-dimensional plastic materials are investigated by the theory of singular points. The unloading wave propagates with a constant velocity, while the propagation velocity of the loading wave is less than that of the unloading wave and the velocity depends upon the stress and the work-hardening. The growth and decay of the amplitude of the waves are also analyzed. The unloading wave propagates with a constant amplitude. The amplitude of the loading wave may grow or decay and the choice between the two depends upon the stress, the work-hardening and whether the wave is compressive or expansive. In the case of growth the amplitude tends to infinity in finite time, that is, the blow time, and the acceleration wave coalesces into a shock wave. In the case of decay the amplitude tends to zero as the time tends to infinity. The propagation velocity, the blow time and the blow distance are calculated and plotted against the strain.