Wave Addition (Superposition) vs. Multiplication (Modulation)
Exploring Sum-to-Product & Product-to-Sum Dualities in Wave Mechanics and Synthesis
2
3
Wave A: Frequency = 2 Hz
Carrier component $y_A = \sin(2\pi f_1 t)$
Wave B: Frequency = 3 Hz
Modulator component $y_B = \sin(2\pi f_2 t)$
Wave A ＋ Wave B (Addition / Superposition)
Sum-to-Product identity: Average carrier wave enveloped by half-difference beat oscillation.
Wave A × Wave B (Multiplication / Ring Modulation)
Product-to-Sum identity: Generates sidebands at 5 Hz (Sum) and 1 Hz (Difference).
Mathematical & Acoustic Mechanism: When Wave A = 2 Hz and Wave B = 3 Hz:
• Addition (Beats):sin(2π · 2t) + sin(2π · 3t) = 2 · sin(2π · 2.5t) · cos(2π · 0.5t) → An average pitch at 2.5 times the base frequency pulses with a 0.5 Hz envelope period (acoustic beating).
• Multiplication (AM Ring Modulation):sin(2π · 2t) · sin(2π · 3t) = 0.5 · cos(2π · 1t) - 0.5 · cos(2π · 5t) → Even though waves of 2 and 3 are multiplied, the original frequencies disappear and new sidebands at 5 Hz (Sum) and 1 Hz (Difference) emerge. This non-linear multiplier underlies audio ring modulators and frequency mixers in quantum telemetry.