1. Formation of the Pattern: When two sine waves are plotted against each other ($X = A \sin(\omega_1 t + \phi)$ and $Y = B \sin(\omega_2 t)$), the resulting path of the electron beam creates complex geometric shapes known as
Lissajous patterns.
2. Shape Characteristics: The shape of the pattern depends on three factors:
• The ratio of the frequencies of the two signals.
• The relative phase difference between the signals.
• The relative amplitudes of the signals.
3. Practical Utility: Lissajous patterns are extremely useful for measuring the frequency ratio or the phase difference between two signals with high precision. For example, if both signals have the same frequency and are in phase, the pattern is a straight diagonal line. If they are $90^\circ$ out of phase, it becomes a circle or ellipse.