# Transverse magnetic waves in parallel planes

* DEFINITION TRANSVERSE MAGNETIC (TM) WAVES OR E WAVES ** :*

In this case, the component of magnetic field vector **H** lies in the plane transverse to the direction of propagation that is there is no component of **H** along the direction of propagation where as component of electric field vector **E** lies along the direction of propagation.

**Derivation transverse magnetic waves between parallel planes:**

As the direction of propagation is assumed as z-direction, therefore,

H_{z }= 0, E_{z} not equals to 0

By substituting Hz = 0 in equation, we get

H_{x} = 0, E_{y} = 0,and E_{x }not equals to 0, H_{y }not equals to 0

Now write wave equation for free space in term of **H**

Ñ^{2}**H** =d

Ñ^{2}**H** =w^{2}με**H** [. . =g^{2}_{g}(jwμ) (σ +jwε)

As σ = 0, then g^{2}g=-w^{2}με]

For the y-component, wave equation will become

d^{2}**H**_{g}/dx^{2} + d^{2}**H**_{g}/dy^{2} + d^{2}**H**_{g}/dz^{2} = – w^{2}με**H**_{y}

As d**H _{y}/d**y = 0 and d

^{2}

**H**

_{g}/dz

^{2}= g

^{2}

_{g}

**H**

_{y}[using assumptions (e) and (f) of article “waves between parallel planes“]

The wave equation becomes

d^{2}**H**_{g}/dx^{2} + g^{2}_{g}**H**_{y} = – w^{2}με**H**_{y}

or d^{2}**H**_{g}/dx^{2} = -(g^{2}_{g} + w^{2}με) **H**_{y}

or d^{2}**H**_{g}/dx^{2} = –** K**^{2}_{g} H_{y} ..(14) (The equation number is continued from article of “transverse electric waves“)

where **K**^{2}_{g} = **Y**^{2}_{g} + w^{2}με

As **H _{y} = H_{y0 }** e

^{–}g

_{g}

^{z}..(15(a))

d^{2}**H**_{g}/dx^{2} =( d^{2}**H**_{g}_{0}/dx^{2} ) e^{–} g_{g}^{z} ..(15(b))

By substituting equations (15) in equation (14), wave equation becomes

d^{2}**H**_{g0}/dx^{2} = –** K**^{2}_{g} H_{y0}

The above equation is a standard differential equation of simple harmonic motion and its solution can be written in the form

**H**_{g0} = A_{3} sin K_{g}x + A_{4} cos K_{g}x

** H**_{g }=( A_{3} sin K_{g}x + A_{4} cos K_{g}x) e^{–} g_{g}^{z} ..(16)

** **(because **H**_{g} = **H**_{g0 }e^{–} g_{g}^{z})

where A_{3 and }A_{4} are arbitrary constants.

Here, the boundary conditions cannot be applied directly to **H**_{g }to determine the constants A_{3 and }A_{4} because, the tangential component of **H** is not zero at the surface of a conductor (H_{tan} ¹ 0). However from equation E_{z} will be obtained in terms of **H**_{g }, and then the boundary conditions would be applied to E_{z}

Differentiate equation w.r.t. x

d**H**_{g}/dx = **K**_{g} (A_{3} cos **K**_{g}x – A_{4} sin **K**_{g}x) e^{–} g_{g}^{z}

From equation

E_{z} = (1/ jwε) d**H**_{y}/d_{x}

Put value of d**H**_{g}/dx in above equation

E_{z} =(** K**_{g}/ jwε) [A_{3} cos **K**_{g}x – A_{4} sin **K**_{g}x) e^{–} g_{g}^{z} ..(17)

**Applying the boundary condition**

that E_{z} = 0 at x = 0 in equation, we get

0 =(** K**_{g}/ jwε) [A_{3} cos **K**_{g}0 – A_{4} sin **K**_{g}x0) e^{–} g_{g}^{z}

Equation reduces to

E_{z} =(** K**_{g}/ jwε) [ – A_{4} sin **K**_{g}x) e^{–} g_{g}^{z} ..(18)

**Now applying boundary condition**

that E_{z} = 0 at x = a in equation, we get

0 =(** K**_{g}/ jwε) [- A_{4} sin **K**_{g}α) e^{–} g_{g}^{z}

Or sin **K**_{g}α = 0

Or **K**_{g}a = mπ

Or **K**_{g} = mπ/α

Equation becomes

E_{z} =(** -K**_{g }A_{4}/ jwε) [ sin (mπ x/α)] e^{–} g_{g}^{z}

E_{z} =** –** mπ x / jwεα A_{4 }sin (mπ x/α)] e^{–} g_{g}^{z} (g_{g} = jb_{g}) ..(19(a))

Substitute equation(19(a)) in 6(c) and integrate

(d**H _{y}/d**

_{g})d

_{x}= jwε (=K

_{g}A

_{4}/ jwε) ( [ sin (mπ x/α)] e

^{–}g

_{g}

^{z})dx

Or **H _{y} = **A

_{4 }(mπ /α) (a/mπ ) cos (mπx /α) e

^{–}g

_{g}

^{z}

Or **H _{y} = **A

_{4 }cos (mπx /α) e

^{-j}

^{b}

_{g}

^{z}..(19(b))

From **E**_{x} **= ****g**_{g}H_{y}

Put equation in above equation

** E**_{x} **= (****g**_{g}/ jwε) A_{4 }cos (mπ x/α) e^{–} g_{g}^{z}

** E**_{x} **= (****b**_{g}/ wε) A_{4 }cos (mπ x/α) e ^{-j}^{b}_{g}^{z} ..(19(c))

Equations (19(a),(b),(c)) represent the expressions for field component for TM waves. In the case of Transverse Magnetic waves there is also the probability of m = 0, as by putting m = 0, in equations (19), some of the fields exist (e.g. E_{x} and H_{y}. Therefore the lowest order mode that can exist in TM waves in the TM_{0} mode.

This is the definition, derivation and discussion of transverse magnetic (TM) waves between parallel planes.