XII. On the resultant of a large number of vibrations of the same pitch and of arbitrary phase

Key Points

• To determine the mathematical distribution and expected amplitude resulting from the superposition of many independent vibrations having the same frequency and random phases.
• Derived analytical probability expressions for the vector sum of numerous sinusoidal vibrations of identical frequency with phases distributed uniformly at random.
• Established the foundational probability distribution for the resultant amplitude of multiple uncorrelated wave sources.
• Showed that the expected total intensity of the combined vibrations equals the sum of the individual component intensities.