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I'm currently in my last semester as a master student and planning to apply for phd degree. I'm going to have a first meeting with my potential phd adviser in a few days and I'm really nervous. All I know is I'm confident about what I've studied and I'm prepared to do the phd study both mentally and physically.

I'm wondering what is the expectation for a professor to accept phd student in general? I've read several research paper of his. However, I only have a general idea about my future research but not a specific topic. Is that ok? What is a decent way to respond when the professor asks about that? What kind of question will usually be asked by the professor during the first meeting. And what kind of question should I ask?

Tips, suggestions, or past experience are all appreciated! Thank you!

update:

The field I'm studying is applied math and I'll doing my phd study in the US.

In most fields in the US, coming to a potential advisor with a proposal isn't necessary. But with an MS, you should have the area you want to study narrowed quite a bit. I don't know specifically about how it works in applied math, but in theoretical math, you generally work out the project details with your advisor given some common interest.

The advisor, if interested in what you are doing can then be more than just someone who looks over you shoulder as you carry on by yourself and can become something of a collaborator. If you are too tightly focused when you first meet, you might hear: "Interesting, but not to me."

But if you want to work at the intersection of, say, analysis and health science, but without anything more specific, then you have a chance to find some common ground with an advisor. The field should be something that you already know about (since you have an MS) and that the potential advisor has previously worked in or has expressed interest in.

If you were a BS graduate instead, very little is actually required (in the US) and you have time to work together with potential advisors in a more general way to develop some common ground for a research problem.

For specifics, just say where you are (intellectually), and what your general interests are. Be prepared to answer about what you have done, and what you have considered.

For completeness, outside the US the answer might be quite different.

I would say it is always better to have at least an idea or an outline of a project by your first interview.

Since you've read the guy's papers, can't you write down a few ideas for extension of the work or to take it in new directions? If you contribute that, you can have a meaningful discussion and also show off the prep.