Chapter one? Derivation of wave equation 1 equation. The condition of definite solution is 1? Thin pole? Or spring water? Longitudinal vibration caused by some external reasons? U(x, t) indicates the deviation of the point at point X from the original position at rest. Suppose the tension in the process of vibration obeys Hooke's law? Try to prove), (txu satisfies the equation? xuExtuxt？ Among them? Is the density of the rod. E is young's modulus. Certificate? Take a shot on the pole? The coordinates of the two ends at rest are x and. xx？ . Now calculate the relative elongation of the rod at time t, and what are the coordinates of the two ends of the rod at time t? ),(); , (the relative elongation of txxuxxtxux is equal to), ()], (()), ([TXXXXXXXXXXXXXXXXXX? The relative elongation of the limit at point X is xu), (t X). According to Hooke's law? Tension), (txT equals), () (), (txuxEtxTx? Where xE is the Young's modulus of point X, let the cross-sectional area of the rod be), (xS acts on the rod segment), (the force at both ends of xxx is xuxsxe) () (xuxxxxetx) (); (). (txx So we get the equation of motion ttuxxsx? )()(? xESutx？ )、(xxxxxESuxx|)(|)(？ Using the differential mean value theorem? Ttuxsx)()(？ x？ XESu () if？ (xs constant? Get 22) tuxedo? (xuxex is proof. 2? When the rod vibrates longitudinally? Suppose (1) endpoints are fixed? (2) Endpoint freedom? (3) The end point is fixed on the elastic support? Try to deduce the corresponding boundary bar jian in these three cases respectively.