Question

do pulsating variable stars, that is, stars that rhythmically expand and contract as a result of thermal instabilities in their atmospheres, generate gravitational waves in this process? why or why not

Answer 1

Yes,  A simple pulsation cycle of pulsating star consists of following events

At one point in the pulsation cycle, a layer of stellar material loses support against the star’s gravity and falls inwards. This inward motion tends to compress the layer, which heats up and becomes more opaque to radiation. Since radiation diffuses more slowly through the layer (as a consequence of its increased opacity), heat builds up beneath it. The pressure rises below the layer, pushing it outwards. As it moves outwards, the layer expands, cools, and becomes more transparent to radiation. Energy can now escape from below the layer, and pressure beneath the layer drops.

Pulsating star doesn't produce gravitational waves as to produce gravitational waves much higher energy is required than a star lose in its pulsation cycle.