Why does a longer string vibrate longer before the damping brings it to rest? While performing string vibration experiments under the boundary condition that both ends are fixed, I observed that the damping of the wave is much larger when the size of the string is smaller and it decreases as the length of the string is increased.
While the string was only 9-10 cm long, under same initial conditions, the vibration lasted less than 3-4 seconds before damping out.
In contrast when I increases the length to 6-7 feet the vibration continued for over 20-25 seconds.
What is the reason behind this?
 A: Unfortunately your question does not specify all of the conditions under which you are observing the effect, so it is not possible to give a definitive answer; however, you might find the following useful.
The elapsed time for the vibration to fade to zero depends on several factors. You might get a better conceptual grasp by considering a comparison with the damping of the motion of a pendulum, as follows:
Unfortunately your question does not specify all of the conditions under which you are observing the effect, so it is not possible to give a definitive answer; however, you might find the following useful.
The elapsed time for the vibration to fade to zero depends on several factors. You might get a better conceptual grasp by considering a comparison with the damping of the motion of a pendulum, as follows:
Frequency. A higher frequency will increase air resistance, all other factors being equal, as the string moves through the air at a higher speed. It will also increase the rate at which the string flexes, which causes energy to be lost as heat in the spring. You don't say in your question whether you have kept the tension constant when increasing the length of the string, or whether you have increased it to maintain the same frequency of vibration, but if the former then that would account for a decreased damping effect.
Amplitude. A vibration with a larger amplitude will take longer to die down. You don't say in your question whether you are taking care to keep the amplitudes the same. 
Mass. A heavier string will have more inertial energy than a lighter one, other factors being constant. In your experiment you have increased the mass of the string by about a factor of 25 by extending its length.
Resonance. A vibrating string will lose energy by vibrating the body to which it is attached through its endpoints. The rate of loss will depend upon the mass, material and geometric properties of the body.
I suspect that the overriding factor in your experiment is the increase of the mass of the string. 
A: The damping was also found to be inversely proportional to cable length. It seems that for certain
length range damping is inversely proportional, while above a critical value the damping is
independent from length,
see references:
Qiu, Y. (2013), ”Investigation of internal damping in carbon fiber and steel cables”,
dissertation, Univ of New Mexico at Albuquerque.
Yamaguchi, H., and Fujino, Yozo. (1987), “Modal damping of flexural oscillation in suspended
cables”, Structural Eng. / Earthquake Eng. Vol. 4. No.2. October pp.413s-421s
