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Reflection of Circular Wave

Reflection of Circular Wave
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2016-12-22 21:49
CircleFunctionTransformationGeometric ConstructionGraphing
① This GeoGebra example dynamically demonstrates how a circular wave (arc-shaped wavefront emitted from a point source) reflects off a straight boundary, visually illustrating the law of reflection. ② Core mathematical concepts include the law of reflection (angle of incidence equals angle of reflection), construction of circular arcs, symmetry of points with respect to a line, and the use of conditional logic (If command) to separate incident and reflected wave regions. ③ Construction details: Point defines the source and boundary points, Line creates the reflecting boundary, CircleArc draws the incident and reflected arc wavefronts, Intersect finds intersection points between arcs and the boundary, and If controls visibility of different regions; the object list includes numerous points (A, B, C, D, E, F, G, H, I, J, K, L, M, N) and auxiliary segments (a_1~a_4, b_1~b_4, c_1~c_4, d_1~d_4, e_1~e_4, f_1, f_2, g_1, g_2) to build dynamic wavefront propagation. ④ Suitable for high school physics or mathematics students learning wave phenomena, reflection laws, and geometric transformations, as well as for introductory university physics demonstrations to understand the geometric essence of wavefront-boundary interactions.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Point(yAxis) → B
  3. Line(A, B) → f
  4. Point(xAxis) → C
  5. Point(xAxis) → D
  6. If(t > 7, Circle[D, t]) → a_2
  7. Intersect(a_2, f, 1) → E
  8. Intersect(a_2, f, 2) → F
  9. CircleArc(D, E, F) → a_3
  10. CircleArc(C, E, F) → a_4
  11. If(t ≤ 7, Circle[C, t]) → a_1
  12. If(t ≤ 8, Circle[C, t - 1]) → b_1
  13. If(t > 8, Circle[D, t - 1]) → b_2
  14. Intersect(b_2, f, 1) → G
  15. Intersect(b_2, f, 2) → H
  16. CircleArc(D, G, H) → b_3
  17. CircleArc(C, G, H) → b_4
  18. If(t ≤ 9, Circle[C, t - 2]) → c_1
  19. If(t ≤ 10, Circle[C, t - 3]) → d_1
  20. If(t ≤ 11, Circle[C, t - 4]) → e_1
  21. If(t ≤ 12, Circle[C, t - 5]) → f_1
  22. If(t ≤ 13, Circle[C, t - 6]) → g_1
  23. If(t > 9, Circle[D, t - 2]) → c_2
  24. If(t > 10, Circle[D, t - 3]) → d_2
  25. If(t > 11, Circle[D, t - 4]) → e_2
  26. If(t > 12, Circle[D, t - 5]) → f_2
  27. If(t > 13, Circle[D, t - 6]) → g_2
  28. Intersect(c_2, f, 1) → I
  29. Intersect(c_2, f, 2) → J
  30. CircleArc(D, I, J) → c_3
  31. CircleArc(C, I, J) → c_4
  32. Intersect(d_2, f, 1) → K
  33. Intersect(d_2, f, 2) → L
  34. CircleArc(D, K, L) → d_3
  35. CircleArc(C, K, L) → d_4
  36. Intersect(e_2, f, 1) → M
  37. Intersect(e_2, f, 2) → N
  38. CircleArc(D, M, N) → e_3
  39. CircleArc(C, M, N) → e_4
  40. Intersect(f_2, f, 1) → O
  41. Intersect(f_2, f, 2) → P
  42. CircleArc(D, O, P) → f_4
  43. CircleArc(C, O, P) → g_3
  44. Intersect(g_2, f, 1) → Q
  45. Intersect(g_2, f, 2) → R
  46. CircleArc(D, Q, R) → d
  47. CircleArc(C, Q, R) → c
  48. Line(S, T) → h