مرکز منطقه ای اطلاع رساني علوم و فناوري - The seismic pulse, an example of wave propagation in a doubly refracting medium

Abstract :

An exact and closed solution is given for the motion produced on the surface of a uniform elastic half-space by the sudden application of a concentrated pressure-pulse at the surface. The time variation of the applied stress is taken as the Heaviside unit function, and its concentration at the origin is such that the integral of the force over the surface is finite. This problem gives an instructive illustration of wave propagation in a doubly refracting medium, since both shear waves and compressional waves are excited, and they travel with different speeds. There is, in addition, the Rayleigh surface wave. For a medium in which the elastic constants $\\lambda$ and $\\mu$ are equal, the vertical component of displacement $\\omega _{o}$ at the surface is given by: $\\omega _{o} = 0$ , $\\tau < frac{1}{\\sqrt {3}}$ , ${\\omega}_{o} = - {Z \\over \\pi\\mu r} \\Bigg\\{ {3 \\over 16} - {\\sqrt{3} \\over 32\\sqrt{{\\tau}^{2} - {1 \\over 4}}} - {\\sqrt{5+3\\sqrt{3}} \\over 32\\sqrt{{3 \\over 4} + {\\sqrt{3} \\over 4} - {\\tau}^{2}}} + {\\sqrt{3\\sqrt{3} - 5} \\over 32\\sqrt{{\\tau}^{2} + {\\sqrt{3} \\over 4} - {3 \\over 4}}} \\Bigg\\}$ , $frac{1}{\\sqrt {3}}< \\tau < 1$ , $\\omega _{o}= - \\frac{Z}{\\pi \\mu r} \\Bigg\\{ \\frac{3}{8} - \\frac{ \\sqrt{5 + 3 \\sqrt{3}}}{16\\sqrt{ \\frac{3}{4} + \\frac{\\sqrt{3}}{4} - \\tau ^{2} }} \\Bigg\\}, 1 < \\tau < \\frac{1}{2} \\sqrt{3+\\sqrt{3}}$ , $\\omega _{o} = - frac{Z}{\\pi\\mu r}frac{3}{8}, \\tau > frac{1}{2}\\sqrt {3 + \\sqrt {3}}$ , where $\\tau = (ct/r)$ , $c$ -shear wave velocity, and $-Z$ is the surface integral of the applied stress. The horizontal component of displacement is obtained similarly in terms of elliptic functions. A discussion is given of the various features of the waves. It is pointed out that in the case of a buried source, an observer on the surface will, under certain circumstances, receive a wave which travels to the surface as an $S$ wave along the ray of total reflection, and from there along the surface as a dif- fracted $P$ wave. An exact expression is given for this diffracted wave. The question of the suitability of automatic computing machines for the solution of pulse propagation problems is also discussed.