Crossing the brachistochrone- The simple harmonic oscillator

The motion of a ship in the sea can be compared to that of the simple harmonic oscillator. In fact the aspect that the natural (initial) period of the wave T is reduced to T΄ implies that as the object (the ship) moves, it pumps energy from the wave. If the wave is described by a simple harmonic oscillator, the equation of motion is written as

and has a solution of the form

where y₀ is the (maximum) amplitude at t=0, and ω₀ is the natural frequency of the oscillator.

The associated energies, the elastic energy E_el, the kinetic energy E_k, and the mechanical (total) energy E_m (which we may also call E₀) of the system are given as follows,

Later on we will include damping and a driving force into the simple harmonic oscillator.

In order to see how we can retrieve the relativistic formula for time dilation, we rewrite the energy equation of the simple harmonic oscillator in the following equivalent way,

Simplifying this expression, we have,

Here we will use the notion of the reference circle as the geometric approximation of an oscillation. If y₀ is the amplitude of the oscillation, and λ₀ is the wavelength, then at a time equal to the period T₀ of the oscillation, a point on the wave moves on a circle whose radius is the amplitude y₀, and whose perimeter is the wavelength λ₀, so that λ₀=2πy₀. Therefore, going back to the previous equation,

where γ_L is Lorentz factor.

The last expression can also be written with respect to the times T₀ and T,

This way we have retrieved the relativistic expression relating the two times, the only difference being that here the two times are T₀ and T, instead of T and T΄, respectively, according to the notation we have used earlier.

Later on we shall see that the reduction of the period of the wave is due to damping.

Notes:

The equations of the simple harmonic oscillator normally refer to a mass m hanging from a spring, where k is the spring constant. But if the ‘spring’ is the sea, and the mass ‘hanging from the spring’ is the ship oscillating in the wave, then we can describe the motion of the ship in the sea with the equation of the simple harmonic oscillator.

Furthermore these equations, as used here, refer to the vertical displacement of the ship,

But since the ship follows the motion of the wave, it is displaced on the vertical axis Δy as much as it is displaced on the horizontal axis Δx. Therefore the displacement on the horizontal axis can be found by replacing y with x in the previous equations.