Since we're intuiting, I'm just imagining something like quickly adding a "D.C. offset" of some given height to the crests and troughs you'd measure by sampling ocean waves.
In fact, I'm not sure I should have quotes around that. Isn't your interlocutor saying a tsunami is literally a direct current of water flowing toward the shore?
I guess it is like a step function, or at least a step function on one side and a really long decay on the other. Is a step function a wave? I’m not sure, my signal processing class was too early in the morning. Maybe it depends on who you ask, mathematicians vs engineers. I’ll go along with the ones that might make a taser or something.