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I would contact the restaurant immediately and ask to speak with the manager. I would then tell him all of the details surrounding the events of last night. Also explain that this is not something that you normally do and apologize for the misunderstanding. This should be enough to clear everything up. Please do not wait to do this though as it will probably only make the situation worse. Also explain that your delay was due to fear as this has never happenned to you before.
I hope this was helpful and informative.
You have taken all of the right steps. Just because they are investigating the matter does not mean that you are going to be charged. Let these people speak to each other and just cooperate should they need any other information from you.
Let me start off by saying this - I don't believe that this is the sort of matter the police really want to concern themselves with, especially for $30. However, if you were charged I would offer to make full restitution plus any kind of fine they might have in mind to resolve the matter.
Sorry if that was not clear. I will give you the exact language :
Where compensation for services is ordinarily paid immediately upon their rendition, as in the case of hotels, restaurants, and comparable establishments, absconding without payment or making payment is prima facie evidence that the services were obtained by deception.
What this means is where food is served and a person leaves without paying there it is presumed that you intended to commit a theft of services. Really, this is just a presumption though and you can prove them wrong, especially in your case.