How did Classical Physics describe oscilliatory motion?

Answer 1

Classical physics could describe oscillator motion perfectly well.
Springs, pendulums (albeit the full unapproximated pendulum comes from the langriangian, which is definitely not what many people learning newtonian physics consider 'classical'), and harmonic motion are all classical concepts.

If you're thinking of the modern physics' application of these oscillator motions to describe our fundamental quantum particles (ie. wave functions, wave-particle duality, probability density), these are actually all mathematical consequences of the Schrodinger equation, which restricts the possible answers to sines, cosines, and other types of orthogonal oscillatory motions.

The only reasons quantum mechanics makes such heavy references to waves and oscillatory motion is because:

1. The math forces it to be that way (sines, cosines, etc).
2. Quantum mechanics and its consequences is confusing enough without having something familiar to liken it to.

Answer 2

Oscillatory motion = motion which continually repeats in time with a fixed period (e.g., the motion of a pendulum in a grandfather clock).