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Like sung notes, string vibrato generally grows in both depth (distance from the original pitch) and speed (frequency of oscillation). But that's controlled by the performer, so the "mathematics" isn't driven by anything nearly as the bow/string/body interaction.

The position of the finger on the string changes (it's a rolling motion), which changes the length of the vibrating string. Shorter string length results in a higher pitch, while longer string length gives a lower pitch. The actual motion may come from the arm, wrist or finger (thus "arm vibrato", "wrist vibrato" and "finger vibrato"), but the physics of it at the contact point is essentially the same.

Unlike vibrato with other instruments, string vibrato doesn't go higher than the the original pitch - only lower, and then back up to pitch. The explanation usually given is this helps hear the top frequency as being stable. Going over the original pitch gives the note a sort of "wobble". It's not clear why this isn't the case for other instruments.

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