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Transverse Magnetic Mode |
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A rectangular waveguide with cross-section  is shown in the figure below |
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The walls of the waveguide are made of ideal conductior and the medium filling the waveguide is ideal dielectric. |
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As a convention and the  -axis is oriented along the broader dimension of the wave guide, and the  -axis is
oriented along the shorter dimension of the waveguide. The  -axis is oriented along the length of the waveguide
and the waveguide is assumed to be infinite length. |
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For transverse magnetic mode, we have  and  . The transverse fields are therefore represented in terms of  components only. |
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The wave equation is to be solved for with appropriate boundary conditions. In Cartesian co-ordinates the wave equation for can be written as |
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 -------- (6.35) |
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The equation can be solved by the separation of variables i.e by assuming that is given as |
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 --------(6.36) |
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The fields are assumed to be of sinosoidal nature with an angular frequency  . |
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Substituting for from 6.36 into 6.35, we get |
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 -------- (6.37) |
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Note that in equation 6.37 the first term is a function of  , the second term is a function of  only, the third term is a function of  only and fourth term is a constant. Since the equation is to be satisfied for every value of  each term in equation 6.37 must be constant i.e |
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| ---------- (6.38) |
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| ---------- (6.39) |
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| ---------- (6.40) |
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 and  are real constants. |
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From the physical understanding of reflection of waves from parallel conducting boundaries we expect a standing wave kind of behaviour in and directions and a travelling wave kind of behaviour in direction. In any case, we expect a wave phenomenon in direction which can be properly represented by putting a negative sign infront of the constant  ,  and  . Instead of negative sign if the positive sign was used the solutions will have real exponential functions which would not represent the wave phenomenon. |
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The equations 6.38, 6.39 and 6.40 can be re-written as |
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| ---------- (6.41) |
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| ---------- (6.42) |
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| ---------- (6.43) |
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These equations are identical to the transmission line equations. |
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The solution to equations 6.41, 6.42 and 6.43 can be appropriately written as |
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---------- (6.44) |
| ---------- (6.45) |
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| ---------- (6.46) |
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where  are arbitrary constants which are to be evaluated by boundary conditions. |
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If we assume that waveguide is of infinite length, we can take only one travelling wave in  -direction. We can |
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then choose  . Substituting for  from 6.44, 6.45 and 6.46 into equation 6.36 the general solution for can be written as |
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---------- (6.47) |
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