Trap: superposition always happens, interference needs coherence
Superposition happens whenever any two waves of the same type meet, coherent or not. What coherence buys you is a pattern that stays fixed, so it can be observed. Two independent lamps do superpose, but their phase relationship changes randomly every 10⁻⁸ s or so, so the maxima and minima reshuffle far faster than the eye or detector can follow and average out to uniform illumination.
What this actually means
So the correct answer to why do two lamps not produce fringes is not that no interference occurs. It is that the interference pattern is not sustained because the sources are incoherent, so no stable pattern is observed.
Attach a timescale if you can. Random phase changes about every 10⁻⁸ s means the pattern changes about 10⁸ times a second, while the eye integrates over about 0.1 s. That is a compelling one-line answer.
The fix is always to derive both beams from one original wave: a single slit before a double slit (division of wavefront) or a half-silvered mirror (division of amplitude).
Two lasers are still not coherent with each other, even though each is individually very coherent. Independent sources cannot be relied on to hold a fixed phase relationship.
Sound is different in practice only because it is easy to drive two speakers from one generator, which makes them coherent by construction.
Writing that incoherent waves do not superpose, rather than that they produce no observable, sustained pattern.
Prove it — watch it be true
- Run the coherent versus randomised-phase demonstration with both sources on.
- In the randomised mode, confirm the instantaneous resultant is still a genuine superposition at every frame.
- Confirm the time-averaged intensity trace is flat, showing why no pattern is observed.