Undergrad What determines if wave components are independent vs coherent?

[ChrisF]

TL;DR

Asking about the physical conditions that determine whether wave components should be treated as independent (sum of squared amplitudes) or coherent (square of summed amplitudes).

I'm working through something and want to make sure I understand the physics.

In a system with three wave components at 120° phase separation, the total energy calculation depends on how we treat them:

If coherent (add amplitudes first, then square):

E = (A₁ + A₂ + A₃)² = 0

If independent (square each, then add):

E = A₁² + A₂² + A₃² = 3/2 = constant

In three-phase electrical systems, we treat the phases as independent — total power is sum of individual powers.

In light interference, we add amplitudes first.

What physical conditions determine which treatment applies? Is it whether the waves share a common source? Whether they're bound vs free? Something else?

Trying to understand the underlying principle.

 

[.Scott]

It's simply whether they are in phase or not.
If you know they are the same frequency and in phase, then they will "constructively interfere".
The method for determining that they are in phase can be anything. For example, two lasers could (in principle) be properly regulated to hold to a specific frequency and phase - and would therefore not be independent.

 

[ChrisF]

Thanks — that makes sense for two waves.


But here's what prompted my question: In three-phase electrical systems, the phases are definitely not independent (same source, fixed 120° relationship). Yet we sum the powers, not the amplitudes.


So is the distinction not "independent vs coherent" but rather "what quantity are we summing"?


With three waves at 120°:

• Amplitude sum = 0 (coherent cancellation)
• Energy sum = A₁² + A₂² + A₃² = 3/2 (constant)

Both are true simultaneously. The system is coherent, but the energy doesn't cancel — it stays constant.


Is this because energy is always additive regardless of phase relationships, while amplitude depends on coherence?

It's simply whether they are in phase or not.
If you know they are the same frequency and in phase, then they will "constructively interfere".
The method for determining that they are in phase can be anything. For example, two lasers could (in principle) be properly regulated to hold to a specific frequency and phase - and would therefore not be independent.

Thanks — that makes sense for two waves.


But here's what prompted my question: In three-phase electrical systems, the phases are definitely not independent (same source, fixed 120° relationship). Yet we sum the powers, not the amplitudes.


So is the distinction not "independent vs coherent" but rather "what quantity are we summing"?


With three waves at 120°:

• Amplitude sum = 0 (coherent cancellation)
• Energy sum = A₁² + A₂² + A₃² = 3/2 (constant)

Both are true simultaneously. The system is coherent, but the energy doesn't cancel — it stays constant.


Is this because energy is always additive regardless of phase relationships, while amplitude depends on coherence?