Explanation

This checkmate pattern happens very often in games, but I couldn't find a good name for it in the English literature. If discussed, it is usually just called "mate on the h-file". In "The art of checkmate" Renaud and Kahn discuss this in the chapter about Lolli's mate after the typical sacrifice of the queen on h7, but it is a different checkmate pattern.

Therefore I am calling this checkmate "high-striker mate", after the attraction often found on funfairs, where you hit with a hammer as strong as you can to ring the bell at the top.
With this metaphor, the checkmating rook or queen is the hammer, and the bell at the top is the checkmated king.
I also like that the name starts with "h" as the h-file involved.

Examples

White can deliver high-striker mate with Rh1#.

This can also happen from behind:

Black has invaded the back-rank with both rooks, and driven the opponent's king to h2. Now Black can give high-striker mate with Rh1#.

Pattern matching

This can happen if there is an open a-file or h-file toward the king. If the file is semi-open, the file can often be openend with a sacrifice.

This is similar to a back-rank mate, but on the a-file or h-file.