1. University of Utah Advanced Electromagnetics Green’s Function Dr. Sai Ananthanarayanan University of Utah Department of Electrical and Computer Engineering www.ece.utah.edu/~psai 1

2. 2 Green’s Function

3. 3 T is the uniform tensile force of the string The string is stationary at the ends, the displacement satisfies the boundary condition

4. 4 Green’s Function Lets first assume that the load applied to the string is concentrated at a point x=x’ Once G(x,x’) is found the displacement u(x) can be obtained by convolving the load F(x) with the green’s function

5. 5 Green’s Function Away from the load at x=x’ the differential equation reduces to the homogeneous form: which has solution of the form

6. 6 Green’s Function Applying the boundary condition A 1 and A 2 are to be determined

7. 7 Green’s Function At x=x’ the displacement of the string must be continuous And hence Green’s function must be continuous at x=x’

8. 8 Green’s Function Substituting

9. 9 Green’s Function

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11. 11 Solution

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13. 13 Closed Form Solution The homogeneous differential equation reduces to:

14. 14 Closed Form Solution Wronskian:

15. 15 Series Form Solution

16. 16 Series Form Solution The amplitude B is such that

17. 17 Series Form Solution

18. 18 Series Form Solution

19. 19 2D Green’s Function Static Fields

20. 20 2D Green’s Function Static Fields

21. 21 Closed Form Solution Representing the Green’s function by normalized single function Fourier series of sine functions that satisfy the BC: Substituting into the equation below

22. 22 Closed Form Solution And applying

23. 23 Closed Form Solution For Homogeneous case with solutions

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