05AB1E, 6 bytes
Çü-0‹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
Explanation:
Ç # Push ASCII values for all characters in the string. ü # Push pairwise array. - # Calculate difference of pairs. 0‹ # Vectorize 1 if negative difference, 0 if positive difference. Ù # Uniquify using right most unique values first.