Microsoft Word - 203 vetted_Final IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jeffry 129 SURFACE EFFECT DUE TO INCIDENT PLANE SURFACE HORIZONTAL WAVES JEFFRY K. Department of Mathematics, Faculty of Mathematics and Natural Science, Hasanuddin University, Jalan Perintis Kemerdekaan, 90245 Makassar, Indonesia. jeffry.kusuma@gmail.com ABSTRACT: Surface effect due to incident plane surface horizontal waves through anisotropic elastic materials is studied. Mathematical formulation over the modeled earth's alluvial valley is transformed into integral equations. These integral equations altogether with the boundary condition and continuity equation are then solved numerically. The solution of these amplification effects over the half circular in homogeneous alluvial valley are shown graphically. ABSTRAK: Kajian tentang kesan permukaan yang disebabkan oleh ombak insiden permukaan mengufuk melalui bahan-bahan elastik anisotropik telah di jalankan. Penggubalan persamaan pengamiran dibuat berdasarkan model lembah aluvium bumi. Persamaan-persamaan ini diselesaikan secara numerikal menggunakan syarat sempadan persamaan selanjar.Penyelesaian kepada kesan pembesaran ke atas model separa bulatan lembah aluvium tak homogen telah digambarkan secara grafik. KEYWORDS: surface effect; surface horizontal wave; inhomogeneous anisotropic materials. 1. INTRODUCTION One of the major concerning in engineering seismology is to understand and explain vibration properties of the soil excited by near earthquakes. Alluvial deposits, often very irregular geometrically, may affect significantly the amplitudes of incident seismic waves.The ground amplification of seismic wave on alluvial valleys have been studied by numerous authors [1-6]. Integral equation formulations have been found to be particularly useful in obtaining numerical solutions to problems of this type. In particular, Wong and Jennings [4] have used singular integral equations to solve the problem of scattering and diffraction of incident surface horizontaly (SH) waves by canyon so farbitrary cross section. Also, Bravo [1] extended the method by considering stratified alluvial deposits. Clements and Larsson [7] extending further these integral formulation techniques by including the case of homogeneous anisotropic materials. In this paper, further extension is being made by including the case of inhomogeneous anisotropic materials. IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jeffry 130 2. GROUND MOTION ON AN ALLUVIAL INHOMOGENEOUS ANISOTROPIC VALLEY Referred to a Cartesian frame �������, let’s consider an anisotropic elastic half space occupying the region �� � 0 as illustrated in Fig. 1. The half space here is divided into two regions in which the first region contains a homogeneous isotropic material with shear moduli � ��� � ��� and the second region contains an inhomogeneous anisotropic material with the shear moduli � ��� � ����������� � ������� � �������. The materials are assumed to adhere rigidly to each other so that the displacement and stress are continuous across the interface boundary between the first and the second regions and the constants in the shear moduli satisfy the symmetry conditions � �Ω� � �Ω� for Ω 1.2. Fig. 1: Incident and reflected waves on the alluvial valley and surrounding half-space. Let ���� and ���� be the displacement in the �� direction in the half space and the valley respectively. For the propagation of horizontally polarizes SH waves, the displacement satisfies the equations of motion � ��� �������� �� ����� � �������� , �1� for the region 1 and ��� �� ����������� � ������� � ������ � ������� � ������������� � ������� � ������ � ��������� , �2� for region 2. Here ���Ω� denotes the density, � denotes the time and repeated Latin subscripts denote summation from to . IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jeffry 131 In view of assuming the form of time dependence as exp �#$��, equation �1� can be reduced to � ��� ��% ����� �� � �����$�% ��� 0, �3� by substituting �������, ��, �� % ������, ��� exp�#$�� . �4� Similarly with the equation . By substituting �������, ��, �� % ������, ��� exp�#$�� , �5� we obtain ��� �� ����������� � ������� � ������ � �% ����� � � �����$��������� � ������� � �������% ��� 0. �6� Our interest is a plane wave of unit amplitude which propagates toward the surface of the elastic half space %*��� exp #$ +� � ��,� � ��,� - , �7� where ,� /���/ sin 4* , ,� /���/ cos 4* , /��� denotes the velocity of the incident waves and 4* denotes the angle of the incident wave. Since %*��� in (7) propagates in the first material, it must satisfy equation (3) so that 7/���8� ������sin�4* � 2������sin4* cos 4* � ������cos�4*����� . �8� IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jeffry 132 Now we consider the case when region 1 and 2 are occupied by the same material. In order to satisfy the traction free condition on �� 0, it is necessary to have a reflected wave of the form %:��� exp #$ +� � ��;� < ��;�- . �9� Thus, if there are no irregularities, the free field solution of the displacement can be written as %>��� %*��� � %:���. �10� The stresses are given by ? ���� � ��� �% ����� , �11� so that the stress ?����� on �� 0 is ?����� @������,� � ��� ��� ,� A exp B#$ +� � ��,� -C � @��� ��� ;� < ��� ��� ;� A exp B#$ +� � ��;�-C . �12� This stress will be zero for all times t if ;� ,�, �13� 1;� 1,� � 2��� ��� ������,� . �14� These equations serves to provide ;� in terms of the unkwown quantities ,�, ,�, ������ and ������. Note that if (9) is substituted into (3) then since it represents a solution to (3) it follows that ������,�� < 2������,�;� � ��� ��� ;�� �����, �15� IIUM Engineering Journal, Vol. 12, No. 6, 2011: Special Issue in Science and Ethics Jeffry 133 and if (13) is used to substitute for 1/;� in (15) and then into (8) so that (14) ensures (9) is a solution to (3) on the assumption that (7) is also solution to (3). Let ;� /D/sin4: and ;� /D/ cos4: where 4: is the angle of the reflection, then tan�4: � ;�;� tan �4* �1 � 2�������/�������tan�4* � , �16� and once 4: has been determined from this equation, the wave speed /Dof the reflected wave may be readily determined from equation /D ;� sin�4: �. To include the influence of the inhomogeneous anisotropic alluvial valley in region 2, the solution for the exterior of the deposit is put in the form % ��� %>��� � %G���, �17� in which %G��� is the displacement due to the diffracted waves. In this region 2, the displacement %��� %:��� will be caused by the refracted waves. 3. INTEGRAL EQUATION Proceeding further as in Clements and Larsson for the region H� with boundary I� and outward pointing normal components J� and J�, the integral equation corresponding to (3) is K%����L, M� N �� ��� �O ����� P J % ���