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

ѲH =d

ѲH =w²μεH                           [.  . =g²_g(jwμ) (σ +jwε)

As σ = 0, then g²g=-w²με]

For the y-component, wave equation will become

d²H_g/dx² + d²H_g/dy² + d²H_g/dz² = – w²μεH_y

As      dH_y/dy = 0 and d²H_g/dz² =  g²_gH_y               [using assumptions (e) and (f) of article “waves between parallel planes“]

The wave equation becomes

d²H_g/dx² + g²_gH_y = – w²μεH_y

or                   d²H_g/dx² = -(g²_g + w²με) H_y

or                   d²H_g/dx² = – K²_g H_y                                                                ..(14) (The equation number is continued from article of “transverse electric waves“)

where                       K²_g = Y²_g + w²με

As                  H_y = H_y0  e^– g_g^z                                                                                    ..(15(a))

d²H_g/dx² =( d²H_g0/dx² ) e^– g_g^z                                                           ..(15(b))

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

d²H_g0/dx² = – K²_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 = A₃ sin K_gx + A₄ cos K_gx

                                   H_g =( A₃ sin K_gx + A₄ cos K_gx) e^– g_g^z                         ..(16)

                                                                                               (because H_g = H_g0 e^– g_g^z)

where A_{3 and} A₄ are arbitrary constants.

Here, the boundary conditions cannot be applied directly to H_g to determine the constants  A_{3 and} A₄ 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

dH_g/dx = K_g (A₃ cos K_gx – A₄ sin K_gx) e^– g_g^z

From equation

E_z = (1/ jwε) dH_y/d_x

Put value of dH_g/dx in above equation

E_z =( K_g/ jwε) [A₃ cos K_gx – A₄ sin K_gx) e^– g_g^z                                 ..(17)

Applying the boundary condition

that  E_z = 0 at x = 0 in equation, we get

0 =( K_g/ jwε) [A₃ cos K_g0 – A₄ sin K_gx0) e^– g_g^z

Equation reduces to

E_z =( K_g/ jwε) [ – A₄ sin K_gx) e^– g_g^z                                                    ..(18)

Now applying boundary condition

that  E_z = 0 at x = a in equation, we get

0 =( K_g/ jwε) [- A₄ sin K_gα) e^– g_g^z

Or                              sin K_gα = 0

Or                              K_ga = mπ

Or                              K_g = mπ/α

Equation becomes

E_z =( -K_g A₄/ jwε) [ sin (mπ x/α)] e^– g_g^z

E_z = – mπ x / jwεα   A₄ sin (mπ x/α)] e^– g_g^z                                                  (g_g = jb_g)          ..(19(a))

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

(dH_y/d_g)d_x = jwε (=K_gA₄/ jwε) ( [ sin (mπ x/α)] e^– g_g^z)dx

Or                  H_y = A₄ (mπ /α) (a/mπ ) cos (mπx /α) e^– g_g^z

Or                  H_y = A₄ cos (mπx /α) e^-jb_g^z                                                               ..(19(b))

From              E_x = g_gH_y

Put equation in above equation

                       E_x = (g_g/ jwε) A₄ cos (mπ x/α) e^– g_g^z

                       E_x = (b_g/ wε) A₄ cos (mπ x/α) e ^-jb_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 TM₀ mode.

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