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Waves and Beams
Published in Rajpal S. Sirohi, Optical Methods of Measurement, 2018
where A is a constant representing the amplitude of the wave at unit distance. This solution represents a spherical wave. A plane wave propagates in a particular direction, whereas a spherical wave is not unidirectional. A quadratic approximation to the spherical wave, in rectangular Cartesian coordinates, is () E(x,y,z)=Azexp(ikz)exp[ik2z(x2+y2)]
Plane Electromagnetic Waves
Published in Philip C. Magnusson, Gerald C. Alexander, Vijai K. Tripathi, Andreas Weisshaar, Transmission Lines and Wave Propagation, 2017
Philip C. Magnusson, Gerald C. Alexander, Vijai K. Tripathi, Andreas Weisshaar
Plane waves are defined as waves that are uniform in direction and magnitude in planes of a stated orientation. If rectangular coordinates are adopted one may begin the analysis by assuming that the E field is (1) everywhere parallel to the X axis, and (2) a function of time and of displacement in the az direction, but (3) not a function of displacement in the ax or ay directions:
On propagation of harmonic plane waves under the Moore–Gibson–Thompson thermoelasticity theory
Published in Waves in Random and Complex Media, 2021
Komal Jangid, Manushi Gupta, Santwana Mukhopadhyay
We observe that the transverse wave is not modified by, nor contributed to, the temperature field of the medium. However, the longitudinal wave is found to be coupled with thermal and mechanical fields. For this reason, we turn our attention on the longitudinal plane wave instead of the transverse wave in the present context. The solutions of the longitudinal plane wave are considered as follows: where γ is the complex wave number, and denote the amplitudes of wave and denotes the non-dimensional angular frequency of the wave. Here represents the component of unit vector in the direction of the displacement, denotes the component of the position vector and represents the component of unit vector normal to the wavefront. We must consider and for physically realistic behavior of wave. Further, denotes the phase velocity of wave, that conform to the longitudinal wave where frequency is denoted by and wavelength is denoted by .
Numerical study on how heterogeneity affects ultrasound high harmonics generation
Published in Nondestructive Testing and Evaluation, 2020
Negar Kamali, Ashkan Mahdavi, Sheng-Wei Chi
A single frequency sinusoidal plane wave, with a frequency of 2 MHz, is applied at one edge of the model and the response waves are measured at the nodes on the opposite edge and then averaged. According to Equations (15) and (16), the calculated maximum element and time-step sizes are 0.3 mm and 0.05 , respectively. Therefore, three levels of mesh refinement with mesh sizes, , and , and corresponding time-step sizes, 0.004 , 0.002 and 0.001 are adopted in the convergence study. It is demonstrated in Figures 4 and 5 that the numerical solutions converge and agree well with the analytical solution, with a maximum error of when mesh size is used.
A high precision crack classification system using multi-layered image processing and deep belief learning
Published in Structure and Infrastructure Engineering, 2020
This filter is used to distinguish cracks from the background in a crack image. A Gabor filter-bank proposed in Jain and Farrokhnia (1991) was implemented in this study. Gabor filtering can be used to estimate the mean and standard deviation of the energy of the filtered image. A Gabor impulse response has a sinusoidal plane wave of orientation and frequency. This can be modulated by a two-dimensional Gaussian envelope as defined by El-Tarhouni, Boubchir, Al-Maadeed, Elbendak, and Bouridane (2016) and Jain and Farrokhnia (1991): and show the frequency and phase of the sinusoidal plane wave, while and represent the space constants of the Gaussian envelop. The Gabor filter-bank used in this research has Gabor filters with a Gaussian kernel function, which is based on sinusoidal plane waves of various orientations from the same Gabor-root filter: where x, y and θ are defined as follows: