after reading through the H2 documentation, I authored this straightforward application to produce a H2 database inside a local directory:

public static void main(String[] args) throws SQLException {

    String path = "C:/Temp/H2/";
    File fpath = new File(path);

    fpath.mkdirs();
    FileUtils.recursiveDelete(fpath);

    String dbName = "tata";
    String connection = "jdbc:h2:file:" + path + dbName;

    Server server = Server.createTcpServer(connection);

    server.start();
    server.stop();

}

The program runs fine, however when I sign in the prospective directory, the database isn't there... (i'm using release 1.3.161)

You have to really access the database, files are produced lazily:

server.start();
DriverManager.getConnection(connection);
server.stop();

Added line in the centre produces tata.h2.db file where expected (examined with 1.3.155).