05AB1E, 11 9 bytes thanks(Thanks to adnanAdnan)
Dg2›iÇü‹Ù Wavy Cases:
0 - Decreasing Wavy
1 - Increasing Wavy
Not Wavy Cases:
[0,1] - Not wavy, initially decreasing, but then has an increase/equality that broke the pattern.
[1,0] - Not wavy, initially increasing, but then has a decrease/equality that broke the pattern
Input String - Not possible to be wavy in the first place due to length.
Explanation:
Dg2›iÇü‹Ù # Full program D # Push 2 copies of input. g2›i # If length is greater than 2. Ç Ç # Push ASCII values for all characters in the string. ü ü # Push pairwise array. ‹ ‹ # Vectorize 1 if negative difference, 0 if positive difference. Ù # Uniquify using right most unique values first. # Else just print the string back, letting them know it's not valid input.