Waveguides are of great importance to both electromagnetics (for example, guiding microwaves into the cooking chamber of a microwave oven to cook food) and the world of acoustics. Waveguides are hollow tubes that guide waves by reflecting them. Without waveguides, waves propagate spherically and decay with range. The waveguides restricts the propagation of the wave to one dimension. These devices can be rectangular or circular in shape. A major area of importance when studying waveguides is boundary conditions. For acoustic waveguides, the boundary conditions are governed by the linearized force equation:
This equation shows that the spacial derivative (gradient) of the pressure (a scalar field) is proportional to density and acceleration. The equation can be used to show that at a rigid boundary where the pressure is maximum, the velocity must be minimum. Calculus tells us that taking the derivative of a function and setting it to zero (and hence the velocity in this case) will yield maximum values. The opposite is also true for a “pressure release” boundary.
We will first consider the case of a rectangular boundaried cavity. A cavity is similar to a waveguide, however the dimensions of a cavity are comparable to each other whereas a waveguide will have one direction that is much longer than the others (to propagate the waves). Applying boundary conditions, it is apparent that if all boundaries are rigid, only standing waves can be contained within the cavity. A pressure equation can be derived from the boundary conditions through substitution into the wave equation. The wave equation will always be satisfied for any kind of wave, including pressure waves. The following equation is substituted into the wave equation:
This results in an equation of three sinusoids in each direction. In this case, the boundary conditions result in cosine (which is maximal at zero (x=0,y=0,z=0)).
Then, from the boundary conditions it must be true that these cosine functions be equal to zero at the boundaries x =Lx, y = Ly, z = Lz.
This leads to solving for the cutoff frequency of the cavity. The wave number (k) is defined as w/c. Solving for w leads to
For waveguides, it is best for the frequency of propagation to be much higher than the cutoff for a decrease in waveguide dispersion. Frequencies below cutoff produce evanescent waves, or waves that die off without propagating.