radius when running straight and is the tread taper (which is the slope of tread in the horizontal direction perpendicular to the track). The path of the wheel set relative to the straight track is defined by a function (), where is the progress along the track. This is sometimes called the tracking error. Provided the direction of motion remains more or less parallel to the rails, the curvature of the path may be related to the second derivative of with respect to distance along the track as approximately It follows that the trajectory along the track is governed by the equation: This is a simple harmonic motion having wavelength: This kinematic analysis implies that trains sway from side to side all the time. In fact, this oscillation is damped out below a critical speed and the ride is correspondingly more comfortable. The kinematic result ignores the forces causing the motion. These may be analyzed using the concept of creep (non-linear) but are somewhat difficult to quantify simply, as they arise from the elastic distortion of the wheel and rail at the regions of contact. These are the subject of frictional contact mechanics; an early presentation that includes these effects in hunting motion analysis was presented by Carter. See Knothe for a historical overview. If the motion is substantially parallel with the rails, the angular displacement of the wheel set is given by: Hence: The angular deflection also follows a simple harmonic motion, which lags behind the side to side motion by a quarter of a cycle. In many systems which are characterised by harmonic motion involving two different states (in this case the axle yaw deflection and the lateral displacement), the quarter cycle lag between the two motions endows the system with the ability to extract energy from the forward motion. This effect is observed in "flutter" of aircraft wings and "shimmy" of road vehicles, as well as hunting of railway vehicles. The kinematic solution derived above describes the motion at the critical speed. In practice, below the critical speed, the lag between the two motions is less than a quarter cycle so that the motion is damped out but, above the critical speed, the lag is greater than a quarter cycle so that the motion is amplified. In order to estimate the inertial forces, it is necessary to express the distance derivatives as time derivatives. This is done