【Swing】 ,  【wave】 ,  【flourish】 ,  【brandish】 ,  【shake】 ,  【thrash】  are comparable when they mean to wield or to handle something so that it moves alternately backward and forward or upward and downward or around and around.

【Swing】 often implies regular oscillations. It may, however, imply continuous rotatory movements.

【Wave】 distinctively implies undulating or fluttering motions without rhythmical regularity and usually additionally implies, according to the nature of the thing waved and the way in which it is waved, a specific intent or purpose (as of signaling, ordering, displaying an emotion, or greeting).

【Flourish】 implies ostentation, triumph, or bravado in swinging or waving something held in the hand (as a weapon, a stick, or a rod).

【Brandish】  stresses menace or threat as  【flourish】 seldom does; otherwise it suggests somewhat similar motions.

【Shake】  (see also  【SHAKE】 2 ) can come very close to  【wave】  or  【flourish】 , but like  【brandish】 it commonly suggests a menacing or threatening or warning intent.

【Thrash】  (see also BEAT 1 ) implies a noisy vigorous swinging suggestive of the motions of a flail in threshing grain.