I know that when plucked as in pizzicato, the violin produces a muted sound. However, I was surprised to find out when I checked with my tuner that if I tune my violin by plucking, it seems somewhat flat when I double check it by bowing! So there's this slight discrepancy between the two. why is that? Is the bowed version correct (which I'm assuming) ? If so then why? also, is it true for any instrument when you play it muted?
Many things can enter in. Bow pressure can force a string out of tune. Try this: tune the open string bowed, then play the string with excessively heavy bow pressure. You'll go out of tune.
Depending on the quality of the instrument, the bridge&soundpost setup, and the phases of Jupiter's moons, you may find that a perfectly tuned (bowed) open string will decay slightly out of tune after you remove the bow. The behavior of a freely oscillating string differs from that of a string driven with a bow.
From a physics standpoint, the resonant frequency of a string under tension, for real-world strings (not those infamous massless ones :-) ), can change with amplitude. Add to that the fact that a plucked string is resonating freely, while a bowed string is actually being caught and released by the bowhairs (at a very high rate), and the math gets well-nigh impossible. I fear I don't have any info as to what the magnitudes of these various effects are. Perhaps someone can chime in w/ some references.
When plucking the string, it vibrates against the resistance of the air. When bowing the string, it vibrates in contact with the intentionally sticky bow. While the bow does a good job continually supplying energy to the string, the "free" movement of the string would be faster than when it swings and sticks.
The reason the bow produces sound is that sliding friction of the bow on the string is lower than static friction. What happens when bowing is that the bow initially starts out grabbing the string and stretching it until the force of the string exceeds the static friction of the bow. At that point, the string will slip and vibrate for one cycle, during which the bow will be applying force in its direction of motion.
After almost a complete cycle, the velocity of the string will be close enough to that of the bow that the bow will again grab the string, thus increasing the amount of force that can be imparted to it, but shortly after that the string will be released and the cycle will repeat.
What will happen essentially is that the bow will grab the string for part of the cycle when it would have been slowing down and make it move slightly faster there than it would have done otherwise. This will have the effect of reducing the time per oscillation cycle, thus raising the pitch.