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From a paper on tunnel design I've been reading: (

In the present application, the solu- tion corresponding to a sinusoidal load in an infinite elastic medium is sought. Since no closed-form solution to this problem exists, a numerical procedure should be used. The procedure involves, first, deriving the solution to the case of a constant pressure applied to a finite strip in an infinite body. The solution for a sinusoidal distribution of loading then can be found by dividing the wavelength into several segments and assuming the pressure on each segment to be constant. In the present case, this procedure is applied to calculate the displacements under a sinusoidal line load. Each wavelength was divided in 10 and 20 segments and a line load of 4, 6, 8 and 10 wavelengths were considered. It was found that the calculated displace- ments became insensitive to the number of wavelengths when the latter exceeded 6, and that 10 segments were enough to represent each wavelength. As a result of this analysis, the vertical displacement under a sinusoidal load may be approximated by

$$u_y = \frac{(3-4\nu)}{16\pi(1-4\nu)G}\sigma L\sin\frac{2\pi x}{L}$$

The authors refer to the plane strain solution to Kelvin's problem before this.

Can you suggest how I would go about deriving this as mentioned in the paper—from going for a finite strip in an infinite body to applying a numerical solution and getting the result for displacement?

Note: $\nu$ poisson's ratio, $L$ Length of strip, $G$ is the shear modulus.

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