This is shown in the terrible straight-to-video sequel, The Little Mermaid 2: Return to Sea. In the film, Ariel's daughter Melody has a rebellious streak just like her mother, and once again the family is put into jeopardy. Only this time, Melody is a human and she wants to become a mermaid. So, in a plot that is definitely not exactly the same as the first movie,
The straight-to-video Disney sequels tend to be lazy attempts to cash in on existing franchises, and we don't really recommend you watch any of them. The one positive thing we'll say on their behalf is at least they typically return the original voice cast. And while we're on the subject, take a look at that picture. What is wrong with Flounder's face?