Waves and Imaginary Numbers

Consider a stagnant pool of water into which a rock is dropped. Waves of fixed wavelength propagate from the center. These waves can be observed

  1. At a fixed position in space as a function of time yielding the time constant, t in units of time or equivantly the frequency w = 1/t
  2. Alternatively, at a fixed instant of time as a function of position in space yielding the wavelength, l, in units of length.
The wave is also described by the maximum wave crest height or the modulus of the sin function also called the wave's amplitude, A. The two measures of the wave's periodicity, l and w can be seen to have an impact on the same feature of the wave, i.e. the phase shift relative to an arbitrary reference point in space and time, x = 0 and t = 0. Then at any point in space or time we can describe the phase angle of the wave in radians as d = 2p(x/l + wt)

and describe the magnitude of the wave as a function of position, x and time t, with A cos(d).

Consider a pendulum as an analogy for the observation of the wave in water as observed from a fixed position in space as a function of time. The pendulum is governed only by the time constant t since its mean position is fixed in space. The kinetic behavior of the pendulum is based on a simple conversion of an energy that is input by the outside action of swinging the pendulum. An ideal pendulum will have no loss. At the highest point of the pendulum swing the inupt energy exists as stored energy, i.e. entirely potential energy (PE). At the lowest point of the swing the input energy exists as kinetic energy, i.e. entirely kinetic energy. The amount of either of these energies in the pendulum at any given point in time is governed by the phase shift relative to the initial point where the energy is entirely potential energy, PE = IE (cos(d))2, where IE is the input energy taken to initially swing the pendulum and d = 2p wt. If we were to consider only the potential energy for the ideal pendulum, the average energy over time stored in the pendulum would be IE/2 since Mean(cos2d) = 1/2. The average kinetic energy would be the same.

At any given instant of time the totaly energy in the system is likely to be partially kinetic and partly potential. The ratios of the two are constantly varying. In order to maintain the input energy for the ideal pendulum it is necessary to describe the pendulum in terms of both the kinetic as well as the potential energies. To Kinetic energy, potential energy is essentially a loss, i.e. KE is lost to PE and visa versa. A mathematical method to describe this relationship is to describe the potential energy by IE {cos(d) + i sin(d)}2, where "i" is the square root of -1 and the second term refers to kinetic energy. It should also be noted that eid = cos(d) + i sin(d). Mean{(eid)2} = 1, so the total energy is maintained in the ideal pendulum at the input energy level.

All waves, including electro magnetic waves, involve an energy conversion process analogous to the pendulum. For electro magnetic waves the input energy is a disturbance in electro-magnetic space such as can be manifested by the change in momentum for a charged particle, e. g. change in direction of propagation of an electron. The larger the magnitude of such a change in momentum the larger the energy of the disturbance in electro magnetic space and the larger the energy (or frequency) of the EM wave that is produced. Through analogy with the pendulum we can express the amplitude of the electric field vector of an electro magnetic wave by E0 eid = E0 {cos(d) + i sin(d)}, where the sin term is related to the magnitude of the magnetic field and d = 2p(x/l + wt).