Two sine waves of the same frequency add constructively or destructively depending on their relative phase

When two sine waves of the same frequency are summed, the result is always another sine wave of the same frequency, but its amplitude depends on the phase difference between the two. When the waves are (nearly) in phase, peaks align with peaks and the sum is louder than either alone — constructive interference. When they are out of phase, peaks align with troughs and the sum is quieter, or cancels entirely — destructive interference. Phase is undetectable for a single sine wave, but the RELATIVE phase between two waves is audible through this loudness change. Destructive interference is the operating principle of active noise cancellation: playing a sine wave of the same frequency but opposite phase as an unwanted noise cancels it. This same phase interaction, when the two frequencies differ slightly instead of matching, becomes beating.

Examples

Two identical 440 Hz sines in phase sum to a 440 Hz sine of double amplitude; the same two exactly out of phase (180 degrees) cancel to silence. Noise-cancelling headphones (Bose, Sennheiser) generate an anti-phase copy of ambient noise to destructively cancel it.

Assessment

Two sine waves of identical frequency and amplitude are summed and the result is silence. What is their phase relationship? Explain why relative phase is audible for two waves but the absolute phase of a single sine wave is not. What technology exploits destructive interference?