How the body vibration on a semi-hollowbody guitar affect the sound through the pickups?

Question

Are this kind of pickups precisely "magnetic pickups"?

If so, the pickups are supposed to be producing sound from the (metallic) string vibration, then how does the body vibration on a semi-hollowbody guitar affect the sound?

Answer 1

Put simply: any time the relative position of the pickup to the metal string changes, there's a current induced in the coil. If the guitar body is vibrating, then the pickups are vibrating as well.
This can be a complicated situation to solve analytically (i.e. with math :-) ), as, e.g., the vibrating body also induces motion in the bridge, which can couple back into the strings. But in any case, there's net motion between the strings and pickups, and thus overtones and whatnot make it down the wire.

Answer 2

These are purely magnetic pickups.

In an mathematically "ideal" electric guitar, the body is completely rigid and doesn't vibrate at all. The only things affecting the vibration of the string are the physical properties of the string itself, how it is picked, and the effect of the magnetic field from the pickup (which is negligibly small).

In solid-body guitars, using dense woods, neck-through bodies, are ways of getting close to that mathematical ideal, where the body has no effect on tone.

Tonewoods are types of wood which are believed to affect the tone, by absorbing vibration, or resonating, in certain way -- although there is some controversy as to how significant this factor is with solid bodies.

In a hollow-body guitar, energy from the string is transmitted through the bridge onto the sound board, and the sound board resonates to create a louder sound. Here the tone of the sound board wood is certainly significant.

So, the string is losing energy through the bridge, and into the sound board -- but that transfer goes in both directions, so the vibration of the sound board transmits through the bridge into the strings. You can prove this by knocking the sound board with your fist, and noticing that the strings sound.

But since the shape and material of the body imparts tone, by transmitting vibration into the body and back to the string, some frequencies are damped, and others are reinforced, meaning that the body shapes the vibration of the string. Hence the vibrations detected by the pickups are different in tone, from the vibrations in a solid body guitar.