Let's do the Wave!

Python 3, 32 bytes

lambda s,n:s+s[s[0]==s[-1]:]*~-n

Concatenates n copies of the string, removing the first character from all copies but the first if the first character matches the last one.

05AB1E, 13 bytes

Uses CP-1252 encoding.

D¬U¤XQi¦}I<×J

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Explanation

-___- and 3 used as input for example.

D              # duplicate input string
               # STACK: "-___-", "-___-"
 ¬U¤X          # push copies of the first and last element of the string
               # STACK: "-___-", "-___-", "-", "-"
     Q         # compare for equality 
               # STACK: "-___-", "-___-", 1
      i¦}      # if true, remove the first char of the copy of the input string
               # STACK: "-___-", "___-" 
         I<    # push input number and decrease by 1
               # STACK: "-___-", "___-", 2
           ×   # repeat the top string this many times
               # STACK: "-___-", "___-___-"
            J  # join with input string
               # STACK: "-___-___-___-"
               # implicitly output

JavaScript (ES6), 47 bytes

f=
(s,n)=>s+s.slice(s[0]==s.slice(-1)).repeat(n-1)
;

<div oninput=o.textContent=n.value&&f(s.value,n.value)><input id=s><input id=n type=number min=1><div id=o>