There might be something wrong in the question. Elastic wave propagats in the plate, and pressure wave propogates only in the acoustic media. So we could not calculate the sound pressure level of a plate. 

Moreover, I'm also wondering what is the background of the question. It is useful for our discussion.

You can calculating sound radiation efficiency of structure from its vibration velocity. This method is described in work "Radiation efficiency as a crucial parameter in transformer noise evaluation" by M. Kozupa and K. Kolber and published in the conference proceeding of Forum Acusticum 2014.

@ Yan-Feng Wang

Could you please tell the difference between elastic wave and pressure wave in acoustic media?

@ Rithesh

The solution to your problem can be found in any standard acoustic or audio and music engineering book. See equation 3.26 in the attached link.

Also, there are alternate variation of the same problem and accordingly the solution. Read standard references for it.

https://books.google.no/books?id=jDeRCSqtev4C&pg=PA201&lpg=PA201&dq=sound+pressure+level+of+a+vibrating+plate+?&source=bl&ots=dGh8yu4501&sig=STmsUUn0gHEb8IbXviJbQnKoLc0&hl=no&sa=X&ved=0ahUKEwj0wvWwr7vKAhWH9HIKHdb-BwkQ6AEIQjAD#v=onepage&q=sound%20pressure%20level%20of%20a%20vibrating%20plate%20%3F&f=false

Hello Nils, if you have a look at my latest paper uploaded (vibroacoustic optimization of sandwich structures with auxetic gradient core) you will find an analytical expression for the radiated sound power. As Bartlomej has indicated, it can be however found in many textbooks.

If the plate is in a baffle, an exact solution of the radiated pressure can be formulated for any point in the half-space using the Rayleigh integral. There are also effective far field approximations. This can be found in textbooks, e.g. "Structure-Borne Sound -

Structural Vibrations and Sound Radiation at Audio Frequencies", Chapter on Sound Radiation from Structures, Authors: Cremer, L., Heckl, M., Petersson, Björn A.T., Springer, 2005.

As a first approximation, express the vibration of the plate as the velocity level (Lv) re 5. 10^-8 m/s. Then the radiated intensity level LI  equals L I= Lv + 10 lg(sigma). sigma is the radiation efficiency. In many cases sigma is not far from unity, again as a first, engineering approximation.

@Vikash Pandey

When we consider the wave propagation in the fluid &solid systems, acoustic wave propagates in the fluid, and elastic wave propagates in the solid parts. Coupling of acoustic and elastic waves exists via the fluid-solid boundary. So there is no sound pressure level in fluid.

While according to the answers provided by the others, it seems that I don't have enough knowledge. I will carefully read the references and improve myself.

@Yan-Feng Wang

Sorry i realize my mistake. I am wanted to know if there is  any analytical solution for the sound pressure level due to the pressure wave that propagates in the  acoustic media(air) due to the vibration of a plate

Thanks everybody. For giving me your valuable inputs .I will go through all the reference provided hopefully I will get the results 

Your question would be calculating the radiation a vibrating plate. as @Bartlomiej Stepien stated you can calculate it using the vibrating velocity. take loudspeaker for example, in piston region (wave length>diaphragm diameter), because the continuity, the air is moving with diaphragm, so the partical velocity and the velocity of diaphragm is identical, thus  the radiation power is Zair*V2. You can testify this by measuring it using both laser doppler vibrometer and microphone. You can also refer to thiele and small's work on 1970s, there are explicit solutions in their work.   such as "direct radiator loudspeaker system analysis, by Small, 1971"