Using a 3-D parallelepiped model of intrinsic Josephson junctions, we calculate the cavity resonance modes of Josephson plasma waves excited by external electric currents. The electromagnetic (EM) wave of the excited Josephson plasma is converted to a THz EM wave at the sample surfaces. The cavity modes accompanied by static phase kinks of the superconducting order parameter have been intensively investigated. The phase kinks induce a spatial modulation of the amplitude of the order parameter around the kinks and decrease the superconducting condensation energy. The Josephson plasma produces a magnetic field in the vacuum in addition to the emitted EM wave. This magnetic energy detemines the orientation of the cavity mode. Taking account of the facts mentioned above, we obtained sharp resonance peaks in the I-V curves and sizable powers of continuous and coherent terahertz wave emission at the cavity resonance. The emission frequencies are inversely proportional to the length of the shorter side of the samples in agreement with experiments. Emission of terahertz electromagnetic (EM) waves from intrinsic Josephson junctions (IJJ) in high temperature superconductors has been extensively studied [1,2,3,4,5,6,7,8]. Recently, Ozyuzer et al. succeeded in detecting strong and continuous emission of terahertz EM waves from mesa-shaped samples of the high temperature superconductor Bi 2 Sr 2 CaCu 2 O 8 (BSCCO) [9]. The general mechanism for the emission is as follows. When an external current is applied along the c-axis, the ac Josephson current in the resistive state excites a cavity resonance mode of Josephson plasma wave in the sample. The excited standing wave of Josephson plasma is converted to a terahertz EM wave at the mesa surfaces and the EM wave is emitted into the vacuum space. However, details of the mechanism have not yet been clarified, although it is important for designing the terahertz EM wave emitters with use of IJJ.Recently, X. Hu and S. Lin [10,11], and A. Koshelev [12] proposed the following new mechanism. When the inductive interaction between the superconducting CuO 2 layers in BSCCO is strong, static kink structures arise in the phase difference of superconducting order parameter between the superconducting layers. The phase kinks induce cavity resonance modes of the Josephson plasma. This is a new dynamic state caused by the non-linear effect special in the IJJ system. In this paper, we first discuss the stability of this new state, and then we investigate the mechanism of the terahertz EM wave emission on the basis of the discussion.For the sample of IJJ, we use a model shown in Fig. 1. In this figure the superconducting CuO 2 layers and the insulating layers in the IJJ are shown in green and light green, respectively. An external electric current is applied in the direction of the z-axis, perpendicular to the layers. The L x , L y , and L z are the sample lengths respectively along the x, y, and z-axes. Now, we derive the equation for the simulation. The superconducting order parameter of t...