While reading through a page on beginner Helioseismology, I occasionally came across the terms p-modes or g-modes, which I know are caused by pressure and gravity. I wanted to know how do satellites used to study the sun detect these solar oscillations.
This field of study is called helioseismology. Solar oscillations can readily be detected using special telescopes on earth, or by instruments in satellites. Here is how.
Solar oscillations manifest themselves as zones of the sun's photosphere (what we think of as the "visible surface" of the sun) which bulge up and sink down like waves in the ocean. A zone which is rising up towards the earth along our line-of-sight to it has a positive component of velocity along that line of sight and a zone which is sinking inwards is moving away from us and has a negative velocity component.
You can deduce those velocities by measuring the doppler shifts created by those line-of-sight velocity variations within those moving zones as functions of position all across the "surface" of the sun. Those doppler shifts are detected with special lens systems and optical sensors, and the raw data is then subjected to processing on powerful computers. the result is then a global map, tracked minute by minute, of all those rising and falling zones.