1. Total field: PEC 𝐸 𝑚||𝑥 𝑖 = 𝐸 𝑚|| 𝑖 cos 𝜃 𝑖 𝛽 𝑥 = 𝛽 𝑖 sin 𝜃 𝑖
= 2 𝐸 𝑚||𝑥 𝑖 sin 𝛽 𝑧 𝑧 cos 𝜔𝑡− 𝛽 𝑥 𝑥− 𝜋 2
𝐸 𝑦 = 2 𝐸 𝑚⊥ 𝑖 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2
= 2 𝐸 𝑚⊥ 𝑖 sin 𝛽 𝑧 𝑧 cos 𝜔𝑡− 𝛽 𝑥 𝑥− 𝜋 2
𝐸 𝑧 = −2 𝐸 𝑚||𝑧 𝑖 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
= −2 𝐸 𝑚||𝑧 𝑖 cos 𝛽 𝑧 𝑧 cos 𝜔𝑡− 𝛽 𝑥 𝑥
𝐸 𝑚||𝑥 𝑖 = 𝐸 𝑚|| 𝑖 cos 𝜃 𝑖
𝛽 𝑥 = 𝛽 𝑖 sin 𝜃 𝑖
𝐸 𝑚||𝑧 𝑖 = 𝐸 𝑚|| 𝑖 sin 𝜃 𝑖
𝛽 𝑧 = 𝛽 𝑖 cos 𝜃 𝑖

2. Reflection by PEC: Total Fields
Parallel Polarization
𝐸 𝑥 = 2 𝐸 𝑚||𝑥 𝑖 sin 𝛽 𝑧 𝑧 𝑒 −𝑗( 𝛽 𝑥 𝑥+ 𝜋 2 )
𝐸 𝑧 = −2 𝐸 𝑚||𝑧 𝑖 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
𝐸 𝑥 =0
sin 𝛽 𝑧 𝑧 =0
cos 𝛽 𝑧 𝑧 =0
𝐸 𝑧 =0
𝜆 𝑧 /2
sin 𝛽 𝑧 𝑧 =0
𝐸 𝑥 =0
𝜆 𝑧 /2
𝐸 𝑧 =0
cos 𝛽 𝑧 𝑧 =0
sin 𝛽 𝑧 𝑧 =0
𝐸 𝑥 =0
𝜆 𝑧 =2𝜋/ 𝛽 𝑧

3. Reflection by PEC: Total Fields
Parallel Polarization
𝐸 𝑥 = 2 𝐸 𝑚||𝑥 𝑖 sin 𝛽 𝑧 𝑧 𝑒 −𝑗( 𝛽 𝑥 𝑥+ 𝜋 2 )
𝐸 𝑧 = −2 𝐸 𝑚||𝑧 𝑖 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
𝑧=− 𝜆 𝑧
sin 𝛽 𝑧 𝑧 =0
cos 𝛽 𝑧 𝑧 =0
𝑧=−3 𝜆 𝑧 /4
𝜆 𝑧 /2
sin 𝛽 𝑧 𝑧 =0
𝑧=− 𝜆 𝑧 /2
𝜆 𝑧 /2
𝑧=− 𝜆 𝑧 /4
cos 𝛽 𝑧 𝑧 =0
sin 𝛽 𝑧 𝑧 =0
𝑧=0
𝜆 𝑧 =2𝜋/ 𝛽 𝑧

4. Reflection by PEC: Total Fields
Effect of incident angle 𝜃 𝑖
𝐸 𝑥 = 2 𝐸 𝑚||𝑥 𝑖 sin 𝛽 𝑧 𝑧 𝑒 −𝑗( 𝛽 𝑥 𝑥+ 𝜋 2 )
𝐸 𝑧 = −2 𝐸 𝑚||𝑧 𝑖 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
𝜆 𝑧 =2𝜋/ 𝛽 𝑧
𝛽 𝑧 = 𝛽 𝑖 cos 𝜃 𝑖
𝛽 𝑖
𝜆 𝑧 = 2𝜋 𝛽 𝑖 cos 𝜃 𝑖 = 𝜆 𝑖 cos 𝜃 𝑖
𝜆 𝑧 ≥ 𝜆 𝑖
Wavefront
𝜆 𝑧 /2
𝜃 𝑖
𝜃 𝑖 =0→ 𝜆 𝑧 = 𝜆 𝑖
𝜆 𝑖 /2
𝜃 𝑖 =90°→ 𝜆 𝑧 =∞
𝜆 𝑥 /2

5. Reflection by PEC: Total Fields
Effect of incident angle 𝜃 𝑖
𝐸 𝑥 = 2 𝐸 𝑚||𝑥 𝑖 sin 𝛽 𝑧 𝑧 𝑒 −𝑗( 𝛽 𝑥 𝑥+ 𝜋 2 )
𝐸 𝑧 = −2 𝐸 𝑚||𝑧 𝑖 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
𝜆 𝑥 =2𝜋/ 𝛽 𝑥
𝛽 𝑥 = 𝛽 𝑖 sin 𝜃 𝑖
𝛽 𝑖
𝜆 𝑥 = 2𝜋 𝛽 𝑖 sin 𝜃 𝑖 = 𝜆 𝑖 sin 𝜃 𝑖
𝜆 𝑥 ≥ 𝜆 𝑖
Wavefront
𝜆 𝑧 /2
𝜃 𝑖
𝜃 𝑖 =0→ 𝜆 𝑥 =∞
𝜆 𝑖 /2
𝜃 𝑖 =90°→ 𝜆 𝑥 = 𝜆 𝑖
𝜆 𝑥 /2

6. Reflection by PEC: Total Fields
Effect of incident angle 𝜃 𝑖
𝐸 𝑥 = 2 𝐸 𝑚||𝑥 𝑖 sin 𝛽 𝑧 𝑧 𝑒 −𝑗( 𝛽 𝑥 𝑥+ 𝜋 2 )
𝐸 𝑧 = −2 𝐸 𝑚||𝑧 𝑖 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
𝛽 𝑖 2 = 𝛽 𝑥 2 + 𝛽 𝑧 2
𝛽 𝑧 = 𝛽 𝑖 cos 𝜃 𝑖
𝛽 𝑥 = 𝛽 𝑖 sin 𝜃 𝑖
𝜆 𝑧 = 𝜆 𝑖 cos 𝜃 𝑖
𝜆 𝑥 = 𝜆 𝑖 sin 𝜃 𝑖
𝛽 𝑥 2 = 𝛽 𝑖 2 − 𝛽 𝑧 2
𝛽 𝑥 = 𝛽 𝑖 2 − 𝛽 𝑧 2
𝛽 𝑖 2 = 𝛽 𝑥 2 + 𝛽 𝑧 2
1 𝜆 𝑥 = 1 𝜆 𝑖 2 − 1 𝜆 𝑧 2
1 𝜆 𝑖 2 = 1 𝜆 𝑥 𝜆 𝑧 2

7. Reflection by PEC: Total Fields
Boundary conditions
𝐸 𝑥 = 2 𝐸 𝑚||𝑥 𝑖 sin 𝛽 𝑧 𝑧 𝑒 −𝑗( 𝛽 𝑥 𝑥+ 𝜋 2 )
𝐸 𝑧 = −2 𝐸 𝑚||𝑧 𝑖 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
𝑛 × 𝐸 =0
𝑛 =− 𝑎 𝑧
𝐸 𝑥 =0
𝑧=0
b.c.
sin 𝛽 𝑧 𝑧 =0
b.c.

8. Reflection by PEC: Total Fields
Boundary conditions
𝐸 𝑥 = 2 𝐸 𝑚||𝑥 𝑖 sin 𝛽 𝑧 𝑧 𝑒 −𝑗( 𝛽 𝑥 𝑥+ 𝜋 2 )
𝐸 𝑧 = −2 𝐸 𝑚||𝑧 𝑖 cos 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥
𝑛 × 𝐸 =0
𝑛 =− 𝑎 𝑧
𝐸 𝑥 =0
𝑧=0
sin 𝛽 𝑧 𝑧 =0

9. Reflection by PEC: Total Fields
Perpendicular Polarization
𝐸 𝑦 = 2 𝐸 𝑚⊥ 𝑖 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2
𝑛 × 𝐸 =0
𝑛 =− 𝑎 𝑧
𝐸 𝑦 =0
𝑧=0
sin 𝛽 𝑧 𝑧 =0

10. Reflection by PEC: Total Fields
Perpendicular Polarization
𝐸 𝑦 = 2 𝐸 𝑚⊥ 𝑖 sin 𝛽 𝑧 𝑧 𝑒 −𝑗 𝛽 𝑥 𝑥+ 𝜋 2
𝑛 × 𝐸 =0
𝑛 =− 𝑎 𝑧
𝐸 𝑦 =0
𝑧=0
sin 𝛽 𝑧 𝑧 =0
We have solved the two-plate waveguide problem.