I can't trace its exact origins, but the phaser and photon torpedo sounds both predate "classic" Trek to at least 1953 where they were used for the Martian war machines (the "manta ray" craft) in George Pal's adaptation of of HG Wells' "War of the Worlds". Remember the sound that rose in pitch as the copper colored craft rose from the pit? Those different pitches eventually became different settings for the phasers. Remember the "thrumming" as green bolts of energy lashed from the "wing tips" of the Martian machines? That sound unchanged was later used for the photon torpedoes.
As I stated, they may even predate "WotW", but if they do, I can't site the source. And just like any studio, Paramount built up a library of effects that could be used in later productions. I distinctly remember the torpedo "twang" in a Martin and Lewis comedy from the 50s. Jerry Lewis is trying to climb into the upper bunk in a train's "sleeper" car. He strikes his skull against the underside of the upper bunk and "KA-THRUMM!!!" with the full echoing resonence.
Sincerely,
Bill