1. Rectangular Waveguide
𝑚=0,1,2,… 𝑛=0,1,2,… 𝑚=𝑛≠0
𝑚=1,2,… 𝑛=1,2, …
𝑇 𝐸 𝑚,𝑛 𝑧
𝑇 𝑀 𝑚,𝑛 𝑧
𝐸 𝑥 = 𝑗𝜔𝜇 𝛽 𝑦 𝛽 𝑥 2 + 𝛽 𝑦 2 𝐴′ cos 𝛽 𝑥 𝑥 sin 𝛽 𝑦 𝑦 𝑒 −𝑗 𝛽 𝑧 𝑧
𝐸 𝑥 =− 𝑗 𝛽 𝑥 𝛽 𝑧 𝛽 𝑥 2 + 𝛽 𝑦 2 𝐴 cos ( 𝛽 𝑥 𝑥) sin 𝛽 𝑦 𝑦 𝑒 −𝑗 𝛽 𝑧 𝑧
𝐸 𝑦 =− 𝑗𝜔𝜇 𝛽 𝑥 𝛽 𝑥 2 + 𝛽 𝑦 2 𝐴′ sin ( 𝛽 𝑥 𝑥) cos 𝛽 𝑦 𝑦 𝑒 −𝑗 𝛽 𝑧 𝑧
𝐸 𝑦 =− 𝑗 𝛽 𝑦 𝛽 𝑧 𝛽 𝑥 2 + 𝛽 𝑦 2 𝐴 sin ( 𝛽 𝑥 𝑥) cos 𝛽 𝑦 𝑦 𝑒 −𝑗 𝛽 𝑧 𝑧
𝐸 𝑧 =0
𝐸 𝑧 =𝐴 sin ( 𝛽 𝑥 𝑥) sin 𝛽 𝑦 𝑦 𝑒 −𝑗 𝛽 𝑧 𝑧
𝐻 𝑥 =− 𝛽 𝑧 𝜔𝜇 𝐸 𝑦
𝐻 𝑥 =− 𝜔𝜀 𝛽 𝑧 𝐸 𝑦
𝐻 𝑦 = 𝛽 𝑧 𝜔𝜇 𝐸 𝑥
𝐻 𝑦 = 𝜔𝜀 𝛽 𝑧 𝐸 𝑥
𝐻 𝑧 =𝐴′ cos ( 𝛽 𝑥 𝑥) cos 𝛽 𝑦 𝑦 𝑒 −𝑗 𝛽 𝑧 𝑧
𝐻 𝑧 =0
𝛽 𝑧 = 𝛽 2 − 𝛽 𝑥 2 + 𝛽 𝑦 2
𝛽 𝑥 = 𝑚𝜋 𝑎
𝛽 𝑦 = 𝑛𝜋 𝑏
𝛽=𝜔 𝜇𝜀

2. Rectangular Waveguide
Dominant mode: T𝐸 1,0 𝑧 (𝑎≥𝑏)
𝑚=1, 𝑛=0
𝛽 𝑥 = 𝜋 𝑎 , 𝛽 𝑦 =0
𝐸 𝑥 =0
𝐻 𝑥 =− 𝛽 𝑧 𝜔𝜇 𝐸 𝑦
𝐸 𝑦 =− 𝑗𝜔𝜇 𝛽 𝑥 𝛽 𝑥 2 + 𝛽 𝑦 2 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧
𝐻 𝑥 =𝑗 𝛽 𝑧 𝑎 𝜋 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧
𝐸 𝑦 =− 𝑗𝜔𝜇 𝛽 𝑥 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧
𝐻 𝑥 =𝑗 𝛽 2 − 𝑎 𝜋 𝑎 𝜋 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧
𝐸 𝑦 =−𝑗𝜔𝜇 𝑎 𝜋 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧
𝐻 𝑦 =0
𝐸 𝑧 =0
𝐻 𝑧 = 𝐴 ′ cos 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧

3. Rectangular Waveguide
Dominant mode: T𝐸 1,0 𝑧 (𝑎≥𝑏)
𝑚=1, 𝑛=0
𝛽 𝑥 = 𝜋 𝑎 , 𝛽 𝑦 =0
𝐸 𝑥 =0
𝐸 𝑦 =−𝑗𝜔𝜇 𝑎 𝜋 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧
𝐸 𝑧 =0
𝐻 𝑥 =𝑗 𝛽 2 − 𝑎 𝜋 𝑎 𝜋 𝐴′ sin 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧
𝐻 𝑦 =0
𝐻 𝑧 = 𝐴 ′ cos 𝜋 𝑎 𝑥 𝑒 −𝑗 𝛽 𝑧 𝑧

4. Rectangular Waveguide
Dominant mode: T𝐸 1,0 𝑧 (𝑎≥𝑏)
𝑚=1, 𝑛=0
𝐸 𝑥 =0
𝐸 𝑦 =𝜔𝜇 𝑎 𝜋 𝐻 0 sin 𝜋 𝑎 𝑥 cos 𝜔𝑡− 𝛽 2 − 𝑎 𝜋 2 𝑧− 𝜋 2
𝐸 𝑧 =0
𝐻 𝑥 = 𝛽 2 − 𝑎 𝜋 𝑎 𝜋 𝐻 0 sin 𝜋 𝑎 𝑥 cos 𝜔𝑡− 𝛽 2 − 𝑎 𝜋 2 𝑧+ 𝜋 2
𝐻 𝑦 =0
𝐻 𝑧 = 𝐻 0 cos 𝜋 𝑎 𝑥 cos 𝜔𝑡− 𝛽 2 − 𝑎 𝜋 2 𝑧

5. Rectangular Waveguide
Cut-off frequency
𝛽 𝑧 = 𝛽 2 − 𝛽 𝑥 2 + 𝛽 𝑦 2 y
𝛽 𝑐 2 = 𝛽 𝑥 2 + 𝛽 𝑦 2 x
𝛽 𝑧 = 𝛽 2 − 𝛽 𝑐 2 b
𝛽 𝑐 =2𝜋 𝑓 𝑐 𝜇𝜀
𝑓 𝑐 = 1 2𝜋 𝜇𝜀 𝛽 𝑥 2 + 𝛽 𝑦 2 = 1 2𝜋 𝜇𝜀 𝑚𝜋 𝑎 𝑛𝜋 𝑏 2 a z

6. Rectangular Waveguide
Transverse E-field: Wave Impedance
𝛽 𝑧 = 𝛽 2 − 𝛽 𝑐 2
𝛽=𝜔 𝜇𝜀
𝐻 𝑥 =− 𝛽 𝑧 𝜔𝜇 𝐸 𝑦
𝑍 𝑤 +𝑧 = 𝐸 𝑥 𝐻 𝑦 =− 𝐸 𝑦 𝐻 𝑥
𝛽 𝑧 =𝛽 1− 𝛽 𝑐 𝛽 2
𝛽 𝑧 =𝛽 1− 𝑓 𝑐 𝑓 2
𝐻 𝑦 = 𝛽 𝑧 𝜔𝜇 𝐸 𝑥
𝑍 𝑤 +𝑧 = 𝜔𝜇 𝛽 𝑧
𝛽 𝑐 = 𝛽 𝑥 2 + 𝛽 𝑦 2
𝑍 𝑤 +𝑧 = 𝜔𝜇 𝛽 1− 𝑓 𝑐 𝑓 2
𝑍 𝑤 +𝑧 = 𝑍 − 𝑓 𝑐 /𝑓 𝑓> 𝑓 𝑐 ∞ 𝑓= 𝑓 𝑐 𝑍 0 𝑗 𝑓 𝑐 /𝑓 2 −1 𝑓< 𝑓 𝑐
𝑍 𝑤 +𝑧 = 𝑍 − 𝑓 𝑐 𝑓 2
𝜔𝜇 𝛽 = 𝜇 𝜀 = 𝑍 0

7. Rectangular Waveguide
Transverse H-field: Wave Impedance
𝛽 𝑧 = 𝛽 2 − 𝛽 𝑐 2
𝛽=𝜔 𝜇𝜀
𝐻 𝑥 =− 𝜔𝜀 𝛽 𝑧 𝐸 𝑦
𝑍 𝑤 +𝑧 = 𝐸 𝑥 𝐻 𝑦 =− 𝐸 𝑦 𝐻 𝑥
𝛽 𝑐 = 𝛽 𝑥 2 + 𝛽 𝑦 2
𝐻 𝑦 = 𝜔𝜀 𝛽 𝑧 𝐸 𝑥
𝑍 𝑤 +𝑧 = 𝛽 𝑧 𝜔𝜀
𝛽 𝑧 =𝛽 1− 𝛽 𝑐 𝛽 2
𝛽 𝑧 =𝛽 1− 𝑓 𝑐 𝑓 2
𝑍 𝑤 +𝑧 = 𝛽 𝜔𝜀 1− 𝑓 𝑐 𝑓 2
𝑍 𝑤 +𝑧 = 𝑍 0 1− 𝑓 𝑐 /𝑓 2 𝑓> 𝑓 𝑐 0 𝑓= 𝑓 𝑐 𝑗 𝑍 𝑓 𝑐 /𝑓 2 −1 𝑓< 𝑓 𝑐
𝑍 𝑤 +𝑧 = 𝑍 0 1− 𝑓 𝑐 𝑓 2
𝛽 𝜔𝜀 = 𝜇 𝜀 = 𝑍 0

8. Rectangular Waveguide
Wave Impedances: 𝑇 𝐸 𝑧 and 𝑇 𝑀 𝑧
𝑍 𝑤 +𝑧 = 𝑍 − 𝑓 𝑐 𝑓 2
𝑇 𝐸 𝑧
𝑍 0 = 𝜔𝜇 𝛽
𝑍 𝑤 +𝑧 = 𝑍 0 1− 𝑓 𝑐 𝑓 2
𝑍 0 = 𝛽 𝜔𝜀
𝑇 𝑀 𝑧