1. Prof. David R. Jackson ECE Dept. Spring 2014 Notes 35 ECE 6341 1

2. Higher-order Steepest-Descent Method Assume This important special case arises when asymptotically evaluating the field along an interface (discussed later in the notes). 2

3. Higher-order SDM (cont.) Along SDP: Hence 3 Note: The variable s is taken as positive after we leave the SP, and negative before we enter the SP.

4. Higher-order SDM (cont.) Define Assume 4 Then

5. Higher-order SDM (cont.) Note: ( s is odd) Hence where Use 5

6. Higher-order SDM (cont.) Recall: 6

7. Higher-order SDM (cont.) Hence We now we need to calculate Note: The leading term of the expansion now behaves as 1 /  3/2. 7

8. Higher-order SDM (cont.) 8

9. Hence We next take two derivatives with respect to s in order to calculate (1st) (2nd) 9

10. Higher-order SDM (cont.) From ( 2nd ): or We then have 10

11. Higher-order SDM (cont.) 11 (3rd) (2nd) or

12. Higher-order SDM (cont.) Hence, 12 Therefore, where

13. Higher-order SDM (cont.) We then have 13

14. Higher-order SDM (cont.) Summary 14

15. Example A line source is located on the interface. 15 Semi-infinite lossy earth

16. Example 16 The field is TM z and also TE y. Semi-infinite lossy earth Consider the line source at a height h above the interface.

17. Example 17 Semi-infinite lossy earth TE y :

18. Example 18 Semi-infinite lossy earth The line source is now at the interface ( h = 0 ).

19. Example (cont.) 19 Simplifying,

20. Example (cont.) Hence Therefore, Now use 20

21. Example (cont.) or 21 Note:

22. Example (cont.) Saddle point: We then identify Note: There are branch points in the complex  plane arising from k y1, but we are ignoring these for a lossy earth ( n 1 is complex). (There are no branch points in the  plane arising from k y0, since the steepest-descent transformation has removed them.) 22

23. Example (cont.) 23 As (unless  1 =  2 ) Hence

24. Example (cont.) Far-field (antenna) radiation pattern: 24

25. Example (cont.) Final far-field radiation pattern: 25 Semi-infinite lossy earth Note: The pattern has a null at the horizon.

26. Example (cont.) 26 Interface field: From the higher-order steepest-descent method, we have:

27. Example (cont.) 27 We then have

28. Example (cont.) 28 We have with

29. Example (cont.) 29 We have

30. Example (cont.) 30 We therefore have with

31. Example (cont.) 31 We then have

32. Example (cont.) 32 We then have where

33. Example (cont.) 33 At the saddle point we have

34. Example (cont.) 34 We then have The field along the interface is then

35. Example (cont.) 35 The field in space (  <  / 2 ) is The field along the interface (  =  / 2 ) is Summary

36. Example (cont.) Space wave Line Source Lateral wave Lossy earth 36

37. Extension to Dipole Dipole Source Space wave Lateral wave Lossy earth 37

38. Important Geophysical Problem Lateral wave TX line source RX line source Space wave Lossy earth The field is asymptotically evaluated for Two types of wave fields are important for large distances:  Space wave  Lateral wave Note: More will be said about these waves in the next chapter on “Radiation Physics of Layered Media.” 38

39. Important Geophysical Problem (cont.) Space wave: Lateral wave: This will be the dominant field for a lossy earth ( k 1 is complex). Note: Amplitude constants have been suppressed here. 39 Lateral wave TX line source RX line source Space wave Lossy earth h1h1 h2h2 