Way - weigh
Way and weigh are sometimes confused in a nautical context, though not commonly in ordinary life. Sailors (and of course academics) regard such selling confusion with horror. Don't be confused.
The confusion in the language of the sea arises, one may consider, from the fact that the spellings of both words, which are homophones, are used in the context of starting a voyage. Sailors begin by weighing anchor - that is, 'feeling its weight' by lifting it - raising it from the sea bed and bringing it onto the vessel. Then they try to get the vessel under way. This is a nautical term, roughly equivalent to the land usage of 'on its way': the root of 'way' is a common Germanic verb which may be translated as 'to journey'. (More technically, a vessel with steerage way has enough speed through the water to enable the use of the rudder to steer her.)
What never properly happens, in ordinary writing, is that a vessel gets under
weigh. That is an erroneous mix of the two idioms. Nautical pedants, on the other hand, hold "a ship is considered to be under weigh only when its anchor has been broken out of the bottom while the ship itself is still stationary" (The Oxford Companion to Ships and the Sea), although they do add that a ship "is under way as soon as it begins to move under its own power.
The above applies also to the compound adjective written as one word. It is always underway, never
- For another homophone confusion, see Way - whey.