why sine wave output from rc phase oscillator?

 

Simple Answer: The differential equation only has one solution, and that is a sine wave.

More Complex Answer: For an oscillator to oscillate at any particular frequency, the feedback loop must have a phase delay of 360 degrees and a gain greater or equal to one at that frequency. In an RC phase oscillator, this is only true at one frequency. A single frequency is a sine wave.

More Info: An actual RC phase oscillator actually has a gain of greater than one so it will continue to grow until the amplifier begins to clip. This effectively drops the gain to exactly one, but will cause distortion due to the nonlinear nature of the clipping.

 

Where a sine wave is important (and many times it is) an automatice gain feature is incorporated in many RC oscillators in the form of a light bulb, which form a variable resistance inside the oscillator that will tweak the gain. The more signal, the brighter the filimant glows, which increases the resistance. When you look around at many sample schematics (as in Wikipedia) you'll see this incorporatied because of its basic simplicity.