Scattering and diffraction of elastic in-plane P- and SV-waves by a surface topography such as an elastic canyon at the surface of a half-space is a classical problem which has been studied by earthquake engineers and strong-motion seismologists for over forty years. The case of out-of-plane SH-waves on the same elastic canyon that is semicircular in shape on the half-space surface is the first such problem that was solved by analytic closed-form solutions over forty years ago by Trifunac. The corresponding case of in-plane P- and SV-waves on the same circular canyon is a much more complicated problem because the in-plane P- and SV-scattered-waves have different wave speeds and together they must have zero normal and shear stresses at the half-space surface. It is not until recently in 2014 that analytic solution for such problem is found by the author in the work of Lee and Liu. This paper uses the technique of Lee and Liu of defining these stress-free scattered waves to solve the problem of the scattering and diffraction of these in-plane waves on an on an almost-circular surface canyon that is arbitrary in shape.

This paper studies the subject on the diffraction of in-plane P-waves in an elastic half-space by arbitrary-shaped canyons using the weighted residual method. It presents a solution for any arbitrary-shaped canyons where the depth of the canyon is approximately half the width of the canyon, such as a semicircle, ellipse, or trapezoid.

Researchers continue to study the effects of scattering and diffraction of waves on two-dimensional canyons in an elastic, isotropic, and homogeneous medium. These studies, which assist researchers to understand earthquake ground motions in and around topographic features, initially, addressed incident SH-waves [

Recently in 2014, Lee and Liu analyzed the harmonic motion induced by an incident P-wave for a two-dimensional diffraction around a semicircular canyon in an elastic half-space using an analytic solution to satisfy the zero-stress boundary conditions [

Arbitrary-shaped with coordinates at the half-space.

This paper expands on Lee and Liu’s new approach for the semicircular canyon replacing it with an arbitrary-shaped canyon with harmonic motion induced by an incident P-wave. In this study, the weighted residual method is applied for the solution of the wave function for the arbitrary-shaped canyon [

The model for the canyon has no restrictions on shape other than the surface of the canyon which must be continuous and defined by a sequential number of points whose polar coordinates have an increasing value of

The half-space is elastic, isotropic, and homogeneous, with the following material properties: Lame constants

Plane longitudinal- (P-) waves enter the half-space at angle

Therefore, using the theory asserted by Lee and Liu [

Boundary conditions on the canyon surface must be satisfied for the free-field and scattered wave potentials. To satisfy these boundary conditions for no force on the canyon surface, the traction components—radial

Traction components.

The stresses due to the scattered waves

On the canyon, the boundary condition of zero stress that must be satisfied on the surface of the arbitrary shape is [

The functions that define the stresses are an infinite summation. Therefore, the traction equations are an infinite summation. An approximate solution would use a finite summation with

The equations for traction, (

The constant terms

The set of

From the above analysis, which determined the coefficient

The free-field displacements are calculated by substituting the incident- and reflected-wave potentials into

Semicircular canyon.

Arbitrary-shaped semicircular canyon

Weighted residual method recreated semicircular canyon matches Lee and Liu’s results

In Figure

The results for the case of the semicircular canyon can also be compared theoretically by matching the boundary condition equations derived from the weighted residual method with those derived from Lee and Liu’s exact closed-form solution [

This new methodology for solving the arbitrary-shaped canyon is applicable to any arbitrary-shaped canyon. This section of the paper will look at an elliptical-shaped canyon and a trapezoidal-shaped canyon. For each shape, the number of equations,

The results for an elliptical canyon of depths

Arbitrary-shaped elliptical canyon.

Ellipse-shaped canyon

Ellipse-shaped canyon

(a) Ellipse-shaped canyon

Ellipse-shaped canyon

Figure

Figure

Figures

In Figure

Figure

The results for the trapezoidal canyon, as shown in Figure

Trapezoidal canyon.

Figure

Trapezoid-shaped canyon

Figure

Trapezoid-shaped canyon

Figure

Trapezoid-shaped canyon

Figure

Trapezoid-shaped canyon

Results of the current research have been presented for various shaped canyons: semicircular, elliptical, and trapezoidal. These results have been presented in Figures

This is the first time that Lee and Liu’s redefined cylindrical-wave function has been used to satisfy the zero-stress boundary condition along the half-space for arbitrary-shaped canyons. The application of the weighted residual method for the solution of the wave function allows for a solution to noncircular arbitrary-shaped canyons.

This analytical approach was applied to a semicircular canyon with

This analytical approach was applied to elliptical-shaped canyons with varying

The elliptical canyon with

This analytical approach was then applied to the trapezoidal canyon. The total displacement amplitudes of both the

At the trapezoidal canyon rims, spikes in the displacement amplitudes were observed. The spikes are consistent with the spikes that were observed for the semicircular and elliptical canyons. These spikes tend to increase in amplitude at higher dimensionless frequencies.

Changing the slope of the trapezoidal canyon walls from 60° to 45° demonstrated that the amplitudes of both the

For the trapezoidal canyon, larger values of

This method provides good results for arbitrary-shaped canyons in which the radius to the canyon surface is approximately equal to

The method used in this paper, which is an expansion of Lee and Liu’s [

The method used in this paper can be applied to canyons of any shape, but convergence of the solutions may be difficult. This is in particular true when the irregular canyon has nonsmooth sharp corners, and when parts of the canyon are too shallow being close to the origin. The sequel to this paper will involve arbitrary-shaped canyons that are much shallower in depth from the half-space surface.

This method can next be applied to arbitrary-shaped cylindrical valleys, where the valley medium is softer or harder than that of the half-space.

This method can also be extended for problems involving three-dimensional elastic-wave propagations using the same concept and methodology but (different) three-dimensional spherical Hankel functions instead of the two-dimensional cylindrical Hankel functions here.

For underground arbitrary-shaped cavities, the above concept of weighted residues can be applied, but the stress-free boundary conditions at the half-space surface will have to be addressed and taken care of, in addition to that at the cavity’s surface.

The two-dimensional diffraction of incident P-waves around an arbitrary-shaped canyon in an elastic half-space is presented in this paper. The scattered wave potentials for the resulting P-waves and S-waves are defined by an infinite series of terms with Hankel functions and only sine terms. Using the zero-stress boundary conditions, a solution is made using the weighted residual method to create a set of simultaneous equations for the unknown coefficients for a finite number of terms for the series.

The solution is applied to the traditional semicircular canyon to verify the results and show convergence with a finite number of terms. The method is then applied to canyons of other shapes to demonstrate the versatility of the methodology. This paper demonstrates that good results can be achieved in deep elliptical (

The method in this paper has versatility and may be applied to other incident waves and to canyons of different shapes. A set of papers have been planned and are ongoing on the same subject area. For example, the work on the solution for the diffraction by shallow arbitrary-shaped canyons has just been completed. Other work planned includes diffraction by arbitrary-shaped fluid and valley media.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank Professor Trifunac for a critical review and suggestion resulting in the improvement of the content and presentation of the paper.