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

Interference between two or more point sources of the same frequency

1. Instructions
   • Click here to start the 2D Waves app and select at the top left Sorgenti: 2 puntiformi (Sources: 2 point sources);
   • Change the positions of the two sources by dragging them with the mouse;
   • Using the corresponding sliders, change the Frequenza della sorgente (source frequency) and the differenza di fase (phase difference) between the sources;
   • To see the interference in the new conditions, restart the phenomenon from the beginning by clicking on the Cancella Onde (clear waves) command
   • You can increase the number of interfering sources by selecting at the top left Sorgenti: 4 puntiformi (Sources: 4 point sources)
     • In this case you may have display problems due to the low brightness of the colours (increase the brillantezza (brightness) with the slider of the same name);
       • You cannot finish the experiment without seeing the three-dimensional view. Select Vista 3D (3D view)
2. Remarks
   • By "default" the program starts in a configuration with sources of the same frequency. Only in this case does a stationary interference pattern appear, in the sense that, at a fixed point in space, the interference conditions do not change in time. A dark point will always stay dark.
3. Related topics: see interference, interference between reflected waves, superposition principle
4. To find out more: see How antennas work, antennas and interference, mathematical aspects of interference, interference of harmonic waves, impedance matching, waves and rainbows

Interference between two point sources of different frequencies

1. Instructions
   • Click here to start the 2D Waves app and select at the top left, in the second drop-down menu, 2 Sorgenti, 2 Freq (2 sources, 2 frequencies);
   • Change (if you like) the positions of the two sources by dragging them with the mouse;
   • Using the corresponding sliders, change Frequenze sorgente 1 and Frequenze sorgente 2 (frequencies of source 1 and source 2);
   • To see the interference in the new conditions, restart the phenomenon from the beginning by clicking on the Cancella Onde (clear waves) command
     • if you have problems due to the low brightness of the colours, increase the brillantezza (brightness) with the slider of the same name
2. Remarks
   • By "default" the program starts in a configuration with sources of the same frequency. Now, however, we can change the frequencies. What we observe is that the interference pattern is no longer stationary but evolves in time: the regions of destructive interference (darkness) move. You can make the interference image stationary again by making the frequencies equal.
3. Related topics: see interference, interference between reflected waves, superposition principle
4. To find out more: see How antennas work, antennas and interference, mathematical aspects of interference, interference of harmonic waves, impedance matching, waves and rainbows, interference of sounds and applications

Interference between waves of different frequencies generated by a single source (beats)

1. Instructions
   • Click here to start the 2D Waves app and select at the top left, in the second drop-down menu, 1 Sorgenti, 2 Freq (1 source, 2 frequencies);
   • Bring the source to the centre of the screen by dragging it with the mouse;
   • Slightly change one of the two source frequencies with the corresponding sliders Frequenze sorgente 1 and Frequenze sorgente 2 (frequencies of source 1 and source 2), so that the two frequencies are almost equal.
   • To see the interference in the new conditions, restart the phenomenon from the beginning by clicking on the Cancella Onde (clear waves) command
     • if you have problems due to the low brightness of the colours, increase the brillantezza (brightness) with the slider of the same name
2. Remarks
   • By "default" the program starts in a configuration with sources of the same frequency. As soon as we slightly change the frequency of one of the two sources, an unexpected phenomenon appears: besides the main oscillation, with a frequency similar to that of the two sources, a sort of slow "pulsation" can be seen, at a much lower frequency than the (almost equal) frequencies of the two sources. In acoustics this phenomenon is called beats.
3. Related topics: see interference, interference between reflected waves, superposition principle
4. To find out more: see How antennas work, antennas and interference, mathematical aspects of interference, interference of harmonic waves, impedance matching, waves and rainbows, interference of sounds and applications, beats