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We derive the collisionless Landau damping in a plasma by satisfying the causal requirement that the susceptibility function of the plasma for time t < 0 should be nil. The causality condition should be satisfied by the susceptibility function of a plasma no matter what equations we employ to describe the plasma. Thus we conclude that the fundamental reason of the collisionless damping can be traced to the causality. As an example, we derive the collisionless damping of ion acoustic wave in a plasma by employing fluid equations.
The axis symmetric analysis method can neither handle initial curved plates nor be used in the optimization of coil shapes because an axis symmetric coil is the only shape to analyze in this method. But the method using some discrete divisions and steps, can overcome these difficulties and show more accurate, reasonable results of temperatures and deflections in flat or curved plates with initial curvature, than those in the axis symmetric analysis method. Traditionally, the coil shape in induction heating is circular shape and it needs the moving process along heating lines. To overcome these, the “long type coil” with some linear parallel coils is proposed. It does not need the moving process along heating lines and reduces the heating process time. The results of experiments are compared with those of simulation.