JavaScript (ES6), 8484 81 bytes
s=>(new Set(st=[...s]).size>2)*(![...s]t.some((c,i)=>c>s[i+1])-![...s]t.some((c,i)=>c<s[i+1])) Assumes the input is all in the same case. Returns 1 for raising wavy, -1 for decreasing wavy, 0 or -0 (both are falsy) for not wavy. Edit: Saved 3 bytes thanks to @RobertHickman.