watson transformation
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2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Wen-Kai Wang ◽  
Guang-Ping Zhu ◽  
Jing-Wei Yin ◽  
Hui Sun

The long-range propagation modes in an acoustic channel under ice are basically caused by supercritical incidence. The energy distribution and transmission loss in the acoustic channel under ice are changed by a scatter in ice. The influence of a slender cylindrical cavity near and parallel to the ice-water interface on the sound propagation is analyzed using Fourier-Bessel series and Sommerfeld-Watson transformation. The research found that the acoustic field presents a beam in the mirror reflection direction at supercritical incidence, and the beam-width is proportional to secant of incident angle; meanwhile, the reflected coefficient is proportional to cosine of incident angle. The reflection coefficient increases with relative depth and Helmholtz number if the incident angle is a constant.


Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 268 ◽  
Author(s):  
Victor Guo ◽  
Michael Schlosser

We provide several new q-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews’ multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the fourth power of a cyclotomic polynomial.


PIERS Online ◽  
2005 ◽  
Vol 1 (6) ◽  
pp. 707-710 ◽  
Author(s):  
Hailong Wang ◽  
Qun Wu ◽  
Xun-jun He ◽  
Jian Wu ◽  
Le-Wei Li

2004 ◽  
Vol 3 ◽  
pp. 75-78 ◽  
Author(s):  
Mao-Kun Li ◽  
Weng Cho Chew

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