*DEFINITION* *TRANSVERSE ELECTRIC (TE) WAVES OR H WAVES** IN PARALLEL PLANES:*

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

**Derivation of transverse electric waves in parallel planes:**

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

E_{z} = 0 and H_{z} is not equal to 0

Now by substituting E_{z} = 0 in equation (8) of article “waves between parallel planes”, we get

E_{x}= 0 and H_{y} = 0 and

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

Now write wave equations for free space in terms of E

Ñ^{2}E =g^{2}_{g}**E**

= -w^{2}με**E** (because g^{2}_{g} = (jwμ) (σ + jwε) As σ =0(from assumption (c) of article “waves between parallel planes” => g^{2}_{g} =-w^{2}με)

Or d^{2}E/dx^{2} + d^{2}E/dy^{2} + d^{2}E/dz^{2} = -w^{2}με**E**

For the y component, the wave equation will become

d^{2}E_{y}/dx^{2} + d^{2}E_{y}/dg ^{2} + d^{2}E_{y}/dz^{2} = -w^{2}με**E _{y}**

also d**E**_{g}/d_{g} = 0 [using assumption (e)]

and E_{g} = E_{g0} e^{–} g_{g}^{z }[using assumption (f)] of article “waves between parallel planes”

Thus d^{2}E_{g}/dz^{2} = g^{2}**E**g^{2}_{g}**E**g^{2}_{g}**E**_{g}

By substituting above values of d**E**_{g}/d_{g} and d^{2}**E**_{y}/dz^{2} in wave equation, we get

d^{2}E_{g}/dx^{2} + g^{2}_{g}**E**_{g}= -w^{2}με**E**_{g}

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

or d^{2}E_{g}/dx^{2} = –**K**^{2}_{g}** E**_{g} ..(9)

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

as E_{g} = E_{g0 }e^{–} g_{g}^{z} ..(10(a))

thus d^{2}E_{g}/dx^{2} = d^{2}E_{g0} e^{–} g_{g}^{z}/dx^{2 }..(10(b))

by substituting equation(10) in equation (9) , wave equation becomes

d^{2}E_{g0}/ dx^{2} = **K**^{2}_{g} E_{g0 }

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

E_{g0} = A_{1} sin **K**_{g}x + A_{2}cos **K**_{g}x

Or E_{g} = (A_{1} sin **K**_{g}x + A_{2} cos **K**_{g}x) e^{–} g_{g}^{z} ..(11)

Where A_{1} and A_{2} are arbitrary constants.

A_{1} and A_{2} can be determined with the help of following **boundary conditions :**

E_{tan} = 0 at the surface of conductor

This implies that

** **E_{y} = 0 at x = 0

E_{y} = 0 at x = a

**Thus apply boundary condition **

That E_{y} = 0 at x = 0

So equation (11)will become

0 = (A_{1} sin **K**_{g}0 + A_{2} cos **K**_{g}0) e^{–} g_{g}^{z}

Thus equation (11) reduces to

E_{y} = A_{1} sin **K**_{g}xe^{–} g_{g}^{z} ..(12)

**Now apply boundary condition **

That E_{y} = 0 at x = a in equation, we get

0 = A_{1} sin **K**_{g}ae^{–} g_{g}^{z}

Or sin**K**_{g}a = 0 as A_{1}¹ 0

Or **K**_{g}a = mπ

Or **K**_{g }= mπ/a, wher m = 1,2,3

(If m = 0, all field components vanish. Its a special case and will be discussed later)

. equation (12)becomes

. . E_{y} = A_{1} sin (mπ/a)e^{–} g_{g}^{z}

E_{y} = A_{1} sin (mπ/a)e^{–}jb^{gz} (. . g_{g} = jb_{g} if α_{g} = 0) ..(13(a))

Put equation (13 a)in (8 d of article “waves between parallel planes”)and integrate

A_{1} sin (mπx/a)e^{–} g_{g}^{z} = jwμ/** K**^{2}_{g} dH_{z}/dx dx

H_{z} = A_{1} **K**^{2}_{g} e^{–} g_{g}^{z} /jwμ sin(mπx/a)/dx

= – A1m^{2}π^{2}/a2 e^{–} g_{g}^{z}/ jwμ cos (mπx/a)/ mπ where **K**^{2}_{g} = m^{2}π^{2}/a^{2}

H_{z} =- A_{1} mπ cos (mπx/a) e^{–} g^{gz}/ jwμ

or H_{z} =- A_{1} mπ cos (mπx/a) e^{–} g^{gz}/ jwμ (g_{g} = jb_{g}) ..(13(b))

Differentiate (13(b))w.r.t. x

dH_{z}/dx = A1m^{2}π^{2} sin (mπx/a) e^{–} g^{gz}/ jwμa^{2}

Substitute above equation in equation

H_{x}=- g_{g} / **K**^{2}_{g} A1 m^{2}π^{2}/ jwμa^{2} sin (mπx/a) e^{–} g_{g}^{z}

H_{x}=- g_{g} m^{2}π^{2}/a^{2} A1 m2π2/jwμa2 sin (mπx/a) e^{–} g_{g}^{z}

H_{x}=- g_{g} A_{1} /jwμ sin (mπx/a) e^{–} g_{g}^{z}

H_{x}=- b_{g} A_{1} /wμ sin (mπx/a) e^{–} g_{g}^{z }( g_{g} = jb_{g}) ..(13(c))

Equation (13(a)), (b), (c)) represent the expressions for field components for TE waves. Each value of m in equations (13) represent a particular field configuration and the wave associated with integer m is designated as TE_{m} wave. The lowest order mode that can exist in this case is TE_{1} mode.

This is the definition, discussion and derivation of transverse electric (TE) waves between parallel planes or plates.