The app controls are labelled in Italian: below, each label is given as it appears in the app, followed by its English meaning.

Cutoff frequency in waveguides

1. Instructions
   • Click here to start the 2D Waves app and select Guide d'onda 1 (waveguides 1) from the menu at the top left
2. Observations
   • You can observe that the speed at which the wave propagates along the waveguide (phase velocity) depends on the width a of the waveguide; this is entirely consistent with the theory: calling z the direction of propagation of the wave and x the direction perpendicular to it, narrow waveguides select larger values of k_x and therefore smaller values of k_z.
   • You can observe a cutoff phenomenon caused by narrow tubes: if k_x coincides with the wave vector k, a cutoff frequency ${\displaystyle f_{t}={\frac {v}{2a}}}$ arises. For small values of a this cutoff frequency increases: when the wave has a frequency lower than the cutoff frequency, it is not transmitted by the waveguide.
   • In this experiment the direction of propagation of the source is parallel to the waveguide, so as to excite the modes in the x direction with low values of n (in practice only the first transverse mode is excited).
3. Other experiments
   • By selecting Guide d'onda 2 (waveguides 2) at the top left, the program changes the frequency of the wave (it lowers it): it is now evident that the cutoff occurs for larger values of a.
   • Try changing the frequency of the wave manually (using the frequenza sorgente 1 (source 1 frequency) slider) to explore the dependence of the wave speed and the onset of the cutoff frequency for different values of the frequency of the incident wave.
4. To find out more visit the pages Waveguides in more depth, interference between reflected waves, dispersion.

Dependence of the cutoff frequency on the order of the transverse modes

1. Instructions
   • Click here to start the 2D Waves app and select Guide d'onda 3 (waveguides 3) from the menu at the top left
2. Observations
   • In this experiment the width of the waveguide is fixed;
   • By means of a system of holes, the direction of propagation of the source has been made different from that of the waveguide (in practice the source now generates a spherical wave);
   • The position of the holes has been chosen so as to generate different modes in the x direction;
   • The changes in phase velocity and the onset of a cutoff frequency are still observed but, since the width of the waveguide is fixed, they are to be attributed to the order n of the transverse modes.
   • In this configuration you can also observe the difference between group velocity (the speed at which the energy of the wave advances along the guide, bouncing between the walls) and phase velocity (the speed at which the wavefronts slide along the guide, greater than the former).
3. Other experiments
   • By selecting Guide d'onda 4 (waveguides 4) at the top left, the program changes the frequency of the wave (it raises it):
   • Try changing the frequency of the wave manually (using the frequenza sorgente 1 (source 1 frequency) slider) to explore the dependence of the wave speed and the onset of the cutoff frequency for the different transverse modes generated at the various frequencies.