This section introduces the concept of kinematic waves, which are waves resulting from the conservation equation. The one-dimensional form of the conservation equation is discussed, and it is noted that a single equation is insufficient to determine both density and velocity. The focus is on the one-dimensional equation where the flux depends only on the density, leading to a first-order quasi-linear partial differential equation. The general solution to this equation is derived using characteristic equations, resulting in an implicit form \( u = f[x - c(u)t] \). The initial value problem is addressed, specifying the initial conditions \( u(x, 0) = f(x) \), and the solution is verified through differentiation and substitution. The validity of the solution is examined, considering the conditions under which the solution may break down. The solution is valid for all time if \( dc(u)/du > 0 \), but if \( dc(u)/du < 0 \), the solution breaks down at \( t_b = 1/|\frac{dc}{du}f'(\xi)|_{max} \). The evolution of the solution in time is discussed, showing that the wave steepens for \( f' < 0 \) and flattens for \( f' > 0 \). An example of traffic flow is provided, where the wave velocity \( c(\rho) \) is derived and the breaking time is calculated. Finally, the section introduces damped waves, where a damping term \( au \) is added to the conservation equation. The characteristic equations and the breaking time for damped waves are derived, showing that the breaking occurs if the initial curve has a sufficiently negative slope.This section introduces the concept of kinematic waves, which are waves resulting from the conservation equation. The one-dimensional form of the conservation equation is discussed, and it is noted that a single equation is insufficient to determine both density and velocity. The focus is on the one-dimensional equation where the flux depends only on the density, leading to a first-order quasi-linear partial differential equation. The general solution to this equation is derived using characteristic equations, resulting in an implicit form \( u = f[x - c(u)t] \). The initial value problem is addressed, specifying the initial conditions \( u(x, 0) = f(x) \), and the solution is verified through differentiation and substitution. The validity of the solution is examined, considering the conditions under which the solution may break down. The solution is valid for all time if \( dc(u)/du > 0 \), but if \( dc(u)/du < 0 \), the solution breaks down at \( t_b = 1/|\frac{dc}{du}f'(\xi)|_{max} \). The evolution of the solution in time is discussed, showing that the wave steepens for \( f' < 0 \) and flattens for \( f' > 0 \). An example of traffic flow is provided, where the wave velocity \( c(\rho) \) is derived and the breaking time is calculated. Finally, the section introduces damped waves, where a damping term \( au \) is added to the conservation equation. The characteristic equations and the breaking time for damped waves are derived, showing that the breaking occurs if the initial curve has a sufficiently negative slope.