Problem description
We are given a two-dimensional area (for simplicity: a rectangle) and we know that a person is missing in this area. We also know that every now and then the person shouts for help, but we do not know how loud this person can shout (we can assume that we know a probability distribution for the loudness of the shouts, the person draws one value from that distribution and uses it throughout) and how often exactly it does so (once a minute versus once an hour versus .., again we can assume to know a probability distribution for the inter-shout times, which are iid). The person is assumed to be stationary (e.g. because it is hurt).
Now there is a ranger tasked with finding that person. To do that the ranger first has to detect an initial shout (depending on how strong the acoustic signal is at the rangers location the ranger might not detect the shout with some probability), and once that has been accomplished, the ranger needs to detect further shouts to localise the person. I am only interested in the first phase until the detection of the first shout.
The problem is to find an optimal path for the ranger to take through the area in order to minimise the average time until the detection of the first shout. Assume that the area is much larger than the largest possible shouting range.
Question
Is this a well-known problem or are there variants which are? Does it (or a variant of it) have a name? Are there any useful references?