The Physics of Interference
This is the reverse of the previous lesson. It shows how adding two waves produces a single wave with a "vibrating" amplitude.
- \(\sin A + \sin B = 2 \sin(\frac{A+B}{2}) \cos(\frac{A-B}{2})\)
- \(\cos A + \cos B = 2 \cos(\frac{A+B}{2}) \cos(\frac{A-B}{2})\)
Worked Examples
Example 1: Adding Waves
Simplify \(\sin(10x) + \sin(2x)\).
- Average: \((10+2)/2 = 6x\). Half-diff: \((10-2)/2 = 4x\).
- Result: \(2 \sin(6x) \cos(4x)\).
Example 2: Visualizing the Result
The result in Example 1 looks like a fast wave (\(\sin 6x\)) whose height is being changed by a slow wave (\(\cos 4x\)). This is exactly how AM (Amplitude Modulation) radio works!
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often have a particle that is in a "Superposition" of two different energy levels. As time passes, the two states add together according to these Sum-to-Product identities. This results in the particle's probability cloud Oscillating back and forth between two locations. This "sloshing" of the wavefunction is exactly what causes an atom to emit light. The frequency of the emitted light is the "difference frequency" (\(A-B\)) found in these identities. All of spectroscopy is based on this simple addition of waves.