The two-source interference equations are not valid for light from an incandescent bulb because this type of light is not monochromatic, it is incoherent, and the observation distance may not be sufficiently large compared to the slit separation.
The two-source interference equations are not valid for light from an incandescent bulb in a double-slit experiment for several reasons. Firstly, the light emitted by an incandescent bulb is not monochromatic. This means that it consists of multiple wavelengths rather than a single wavelength, causing multiple overlapping interference patterns which obscure the clear fringe pattern expected from monochromatic light.
Secondly, incandescent bulbs produce incoherent light. For interference patterns to be observed, the light sources need to be coherent, where the waves are in phase or have a definite phase relationship. Incoherent light sources have random phase relationships that prevent a stable interference pattern from developing on the screen.
Finally, for the two-source interference pattern to be properly observed, the observation distance should be much larger than the separation distance between the slits. This ensures that the slight differences in path length that lead to constructive and destructive interference can manifest clearly on the observing screen. If the observation distance is similar to or smaller than the separation between the sources, the expected interference pattern may not resolve properly.