**Subject :**Microwave Engineering

## Solutions of Wave equations in Rectngular Waveguide

**Introduction : -**

We are analyzing the wave euations and solving these equations for rectnagular waveguides in this section.

**Solution of wave equations :**

A rectangular waveguide system is shown in the following figure :-

**Fig : Rectangular waveguide
**

The electric and magnetic wave equations in frequency domain are : -

................. (1)

These are called the vector wave equation.

Rectangular coordinates are the usual right-hand system. The rectangular components of E or H satisfy the complex scalar wave equation or Helmholtz equation :

................(2)

The Helmholtz equation in rectangular coordinates is :

.................(3)

This is a linear and inhomogeneous partial differential equation in three dimensions. By the method of separation of variables, the solution is assumed in the form of

.............(4)

where, X(x) = a function of the x coordinate only

Y (y) = a function of the y coordinate only

Z(z) = a function of the z coordinate only

Substitution of Eq.(4) in Eq. (3) and division of the resultant by Eq. (4) yield :

............(5)

Since the sum of the three terms on the left-hand side is a constant and each term is independently variable, it follows that each term must be equal to a constant.

Let the three terms be k_{x}^{2} ,k_{y}^{2} and k_{z}^{2} respectively; then the separation equation is given by:

...............(6)