When dealing with interference patterns, we add intensities (or power) rather than amplitudes because intensity is a measure of the energy or power carried by a wave. Intensity is proportional to the square of the amplitude of a wave.
Interference occurs when two or more waves overlap and combine with each other. When waves interfere constructively, their amplitudes add up, resulting in a larger amplitude and, consequently, a higher intensity. On the other hand, when waves interfere destructively, their amplitudes can cancel each other out, resulting in a lower amplitude and a lower intensity.
When we add the amplitudes of two waves, the resulting amplitude depends on the phase relationship between the waves. If the waves are perfectly in phase, their amplitudes add up, but if they are out of phase, their amplitudes subtract from each other. However, the intensity of a wave is proportional to the square of its amplitude. Therefore, when adding intensities, we square the amplitudes and then add them together.
Mathematically, if we have two waves with amplitudes A1 and A2, the resulting intensity I is calculated as:
I = I1 + I2 = (A1^2) + (A2^2),
where I1 and I2 are the individual intensities.
By adding the intensities, we properly account for the energy carried by the waves and obtain the correct result when dealing with interference patterns.