A more general model equation of nonlinear Rayleigh waves and their quasilinear solutions

2016 ◽  
Vol 30 (08) ◽  
pp. 1650096 ◽  
Author(s):  
Shuzeng Zhang ◽  
Xiongbing Li ◽  
Hyunjo Jeong
Keyword(s):  
Rayleigh Waves ◽  
General Equation ◽  
Wave Motion ◽  
Numerical Solutions ◽  
Second Harmonic ◽  
Approximate Equation ◽  
Finite Amplitude ◽  
Sound Beam ◽  
Quasilinear Theory ◽  
Sound Beams

A more general two-dimensional wave motion equation with consideration of attenuation and nonlinearity is proposed to describe propagating nonlinear Rayleigh waves of finite amplitude. Based on the quasilinear theory, the numerical solutions for the sound beams of fundamental and second harmonic waves are constructed with Green’s function method. Compared with solutions from the parabolic approximate equation, results from the general equation have more accuracy in both the near distance of the propagation direction and the far distance of the transverse direction, as quasiplane waves are used and non-paraxial Green’s functions are obtained. It is more effective to obtain the nonlinear Rayleigh sound beam distributions accurately with the proposed general equation and solutions. Brief consideration is given to the measurement of nonlinear parameter using nonlinear Rayleigh waves.

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

Interrogating the health condition of rails using the narrowband Rayleigh waves emitted by an innovative design of non-contact laser transduction system

2020 ◽  
pp. 147592172096760
Author(s):  
Faeez Masurkar ◽  
Kim Ming Ng ◽  
Peter W Tse ◽  
Nitesh P Yelve
Keyword(s):  
Health Status ◽  
Optical System ◽  
Rayleigh Waves ◽  
Rail Steel ◽  
Second Harmonic ◽  
Laser Doppler Vibrometer ◽  
Health Condition ◽  
Innovative Design ◽  
Scanning Laser Doppler Vibrometer ◽  
Contact Experiment

The article reports an innovative optical system that is designed to interrogate the health condition of macroscopically intact rail specimens by measuring its inherent nonlinearity using the narrowband Rayleigh waves. A line-arrayed pattern is developed through the optical system that generates narrowband Rayleigh waves with high power on the surface of the rail. As a result of lattice-anharmonicity, a second harmonic is produced in the wave that is sensed by a scanning laser Doppler vibrometer. The spectral amplitudes of the first and generated second harmonics are used to calculate the inherent nonlinearity using an amplitude-based nonlinearity equation. These measurements are carried out on the head, web, and foot of the rail. The performance of the non-contact experiment is also compared with that of a contact experiment carried out using wedge transducers. The experimentally evaluated nonlinearity of the rail steel is further compared with that obtained using a physics-based nonlinearity equation that relies on the higher-order elastic constants. Agreement of the results shows that the new optical system is effective in generating Rayleigh waves in rails and thereby measuring the inherent nonlinearity of the rail track. The estimation of inherent nonlinearity may help in diagnosing the health status of the macroscopically intact rail specimens in terms of their microstructural consistency and level of dissolved impurities before fixing them on a track.

Numerical simulation of two-dimensional compressible convection

1975 ◽  
Vol 70 (4) ◽  
pp. 689-703 ◽  
Author(s):  
Eric Graham
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Fluid Velocity ◽  
Critical Rayleigh Number ◽  
Finite Amplitude ◽  
Two Dimensions ◽  
Onset Of Convection ◽  
Local Sound ◽  
Constant Viscosity ◽  
Convection Problem

A procedure for obtaining numerical solutions to the equations describing thermal convection in a compressible fluid is outlined. The method is applied to the case of a perfect gas with constant viscosity and thermal conductivity. The fluid is considered to be confined in a rectangular region by fixed slippery boundaries and motions are restricted to two dimensions. The upper and lower boundaries are maintained at fixed temperatures and the side boundaries are thermally insulating. The resulting convection problem can be characterized by six dimension-less parameters. The onset of convection has been studied both by obtaining solutions to the nonlinear equations in the neighbourhood of the critical Rayleigh number Rc and by solving the linear stability problem. Solutions have been obtained for values of the Rayleigh number up to 100Rc and for pressure variations of a factor of 300 within the fluid. In some cases the fluid velocity is comparable to the local sound speed. The Nusselt number increases with decreasing Prandtl number for moderate values of the depth parameter. Steady finite amplitude solutions have been found in all the cases considered. As the horizontal dimension A of the rectangle is increased, the length of time needed to reach a steady state also increases. For large values of A the solution consists of a number of rolls. Even for small values of A, no solutions have been found where one roll is vertically above another.

scholarly journals Comparative evaluation of in situ stress monitoring with Rayleigh waves

2018 ◽  
Vol 18 (1) ◽  
pp. 205-215 ◽  
Author(s):  
James Martin Hughes ◽  
James Vidler ◽  
Ching-Tai Ng ◽  
Aditya Khanna ◽  
Munawwar Mohabuth ◽  
...  
Keyword(s):  
Rayleigh Wave ◽  
Harmonic Generation ◽  
Health Monitoring ◽  
Rayleigh Waves ◽  
Second Harmonic ◽  
Monitoring Systems ◽  
Stress Monitoring ◽  
Bulk Waves ◽  
Health Monitoring Systems

The in situ monitoring of stresses provides a crucial input for residual life prognosis and is an integral part of structural health monitoring systems. Stress monitoring is generally achieved by utilising the acoustoelastic effect, which relates the speed of elastic waves in a solid, typically longitudinal and shear waves, to the stress state. A major shortcoming of methods based on the acoustoelastic effect is their poor sensitivity. Another shortcoming of acoustoelastic methods is associated with the rapid attenuation of bulk waves in the propagation medium, requiring the use of dense sensor networks. The purpose of this article is twofold: to demonstrate the application of Rayleigh (guided) waves rather than bulk waves towards stress monitoring based on acoustoelasticity, and to propose a new method for stress monitoring based on the rate of accumulation of the second harmonic of large-amplitude Rayleigh waves. An experimental study is conducted using the cross-correlation signal processing technique to increase the accuracy of determining Rayleigh wave speeds when compared with traditional methods. This demonstrates the feasibility of Rayleigh wave–based acoustoelastic structural health monitoring systems, which could easily be integrated with existing sensor networks. Second harmonic generation is then investigated to demonstrate the sensitivity of higher order harmonics to stress-induced nonlinearities. The outcomes of this study demonstrate that the sensitivity of the new second harmonic generation method is several orders of magnitude greater than the acoustoelastic method, making the proposed method more suitable for development for online stress monitoring of in-service structures.

Propagation and interaction of two collinear finite amplitude sound beams

1987 ◽  
Vol 82 (S1) ◽  
pp. S12-S12
Author(s):  
Jacqueline Naze Tjøtta ◽  
Sigve Tjøtta ◽  
Erlend H. Vefring
Keyword(s):  
Finite Amplitude ◽  
Sound Beams

scholarly journals KINEMATICS AND RETURN FLOW IN A CLOSED WAVE FLUME

1988 ◽  
Vol 1 (21) ◽  
pp. 31 ◽  
Author(s):  
Jerald D. Ramsden ◽  
John H. Nath
Keyword(s):  
Water Waves ◽  
Water Surface ◽  
Large Scale ◽  
Wave Motion ◽  
Finite Amplitude ◽  
Return Flow ◽  
Order Stream ◽  
Wave Flume ◽  
Seventh Order ◽  
Surface Measurements

Stokes (1847) showed that finite amplitude progressing waves cause a net drift of fluid, in the direction of wave motion, which occurs in the upper portion of the water column. In a closed wave flume this drift must be accompanied by a return flow toward the wave generator to satisfy the conservation of mass. This study presents Eulerian velocity and water surface measurements soon after the onset of wave motion from 12 locations in a large scale flume. Waves with .67 < kh < 2.29 and .09 < H/h < .39 were produced in a water depth of 3.5 meters. Superimposing the return flow theory of Kim (1984) with seventh order stream function theory is shown to improve the velocity predictions. The measured return flows are a function of time and depth and agree with Kim's theory as a first approximation. The mean water surface set-down agrees with the theory of Brevik (1979) except for the nearly deep water waves.

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