Expressing wish using past perfect and simple past

What's the difference between these two sentences? One uses past perfect, the other uses simple past. Which is more appropriate for wishing that something really happened in the past?

I wish I had studied English.

I wish I studied English.

Am I required to specify an event when using the past perfect? E.g.

I wish I had studied English before I came to the United States.

The answer from @WS2 is good as far as it goes.

But "I wish I studied English" does work. Consider

I wish I worked at IBM.

This means I wish I were employed at IBM.

So with the English study, the sentence without the "had" means I wish I were enrolled in an English class or I wish I had an English teacher to work with. (Self-study would also be an option, of course.)

Granted, "I wish I had studied" (in the past) is more commonly encountered.