Over the past decade, significant attention has been devoted to computing and understanding the exact traveling wave solutions (TWS) of the Navier-Stokes equations (NSE). To better understand the linear and nonlinear mechanisms in the TWS, we consider a low-order approximation of the TWS in terms of a weighted sum of resolvent modes. The resolvent modes represent the most amplified forcing and response modes by the linear mechanisms in the NSE and the weights represent the scaling influence of the nonlinear terms. We show that only a few resolvent modes are sufficient to capture most of the dynamics of the TWS in channel and pipe flows. For the most energetic Fourier modes in the wall-parallel directions, it is shown that the first few most amplified resolvent modes capture approximately 90% of the energy of the velocity field. This illustrates the integral role of linear amplification mechanisms in the NSE in shaping the wall-normal profile of the response. In addition, we show that approximately less than 30% of the nonlinear terms in the NSE is captured by the corresponding resolvent forcing modes. Therefore, a relatively small portion of the Reynolds-stress gradient is required for sustaining the velocity fluctuations.