During wave propagation through conductor, why is the current density out of phase with electric field ? When the oscillating electric field applies a driving force on the free electrons, they will start oscillating. Then the current density should be in phase with electric field , isn't it ?
The effect is identical to that of a harmonic oscillator. The electric field accelerates the electrons similar to how the spring accelerates the mass at the end. In the harmonic oscillator, too, the velocity is the highest when the force is lowest, i.e. when the displacement is zero.
If I have to guess at the origin of your confusion, you are trying to apply Ohms law $J=\sigma E$, but this applies in a different situation than what you describe. The reason is as follows. Electrons cannot move freely though the conductor, but they will collide with the atoms, meaning they cannot accelerate indefinitely. How often they collide gives the value of $\sigma$ in Ohms law. If you create an electric field that oscillates very rapidly, the electrons will not accelerate to high speeds, and will not travel for great lengths, so these collision effects will not be large. In this case, the behaviour will be similar to the harmonic oscillator, rather than to what Ohms law describes.