: How to write in an academic way and sound professional? For example, how could I write the following sentences in an academic way? After I have contacted Dr. .., he has accepted to be on

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First, the obvious:

Use formal language: no contractions, no slang, no casual speech forms
Avoid typos and grammatical errors like the plague: one misplaces apostrophe can make even the most intellectual statement sound dumb
Be complete and concise: don't say anything you don't need to say; do say everything you do need to say

In general, assume you are writing to busy professionals who are well versed in whatever you are discussing, and write so they can get through your text without confusion of distraction. If they can do that, they will tend to see you as professional in your own right.
On a more subtle level, be direct and declarative; don't beat around the bush with social niceties, but get to the meat of the matter. For example, Where you wrote this:

After I have contacted Dr. _, he has accepted to be on my supervision
panel, he told me that he can contact you and ask you if you are
interesting in being on my supervision panel. Then, he informed me
that you have you are interested. Thus, I would like to thank you for
accepting to be on my supervisor panel.

I would say:

Dr. _ informed me that that you are willing to be on my supervision
panel. Thank you, I am honored you would do that.

There's no need to repeat the process by which this happened, which the person you are writing to (Dr. X) already knows. There is certainly no need to tell X what X said to _. And there is no need to beat around the bush with empty filler words like 'I would like to'. Be active: don't 'like to thank', don't 'would like to thank'; just 'thank' and move forward.

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Generally, when I had to write grant proposals and other professional scientific materials for my administration, I followed a few guidelines:
No contractions or slang, and try especially hard to avoid typos and grammatical errors.
For example, you seem to have a grammatical error in "if you are interesting" ("if you are interested"). You also appear to have placed a comma where you should have put a semicolon ("...on my supervision panel; he told me...") Even where you've written "After I have contacted..." that should probably be "After I contacted." Overall, the grammar of this piece of writing needs a bit of work, unfortunately.
Mistakes reduce the professional sound of your writing, and while you should avoid this in any piece of writing, you should especially be careful in professional writing. My research group had a grant proposal sent back for revision due to a mistake in one of our subheadings.
Speak in the third person, and use passive voice.
Instead of saying, "I measured the properties of the chemical," you would say, "The properties of the chemical were measured." This seems very counterintuitive, because active voice is highly encouraged over passive voice in narrative writing, but in professional and technical writing, passive voice is actually the way to go most of the time.
Use compound sentence structure, and avoid fragments and short sentences.
In narrative writing, it's perfectly fine to use fragments and shorter, choppier sentences, and many people use it as an integral part of their style. However, in professional writing, you want all of your sentences to be clear, complex and fully fleshed out, with commas and connectors appropriately placed. Instead of saying:

The substance was placed in the accelerator. The accelerator was turned on. The reaction was observed. The results were recorded.

You would probably revise it to say:

The substance was placed in the accelerator, after which the accelerator was turned on; following this, the reaction was observed and the results were recorded.

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