Here's a visual of your thoughts from the Fukushima event:

https://m.youtube.com/watch?v=-VcWF8dIDj4

Japan, Tsunami. Coast Guard ship rides over the tsunami waves. 日本 - 津波 4.1M views · 14 years ago

Neat video. It seems to capture the middle between "super high coastal wave" and "shallow but long duration wave in deep water". That's what you'd get with coastal waters several tens of meters deep.