Foundation · Explanatory preview

Fields, continuous waves, and harmonic functions

A field assigns physical values across continuous space. Harmonic waves propagate oscillations with phase kx - omega t, where optical intensity reflects the time average of cos^2 equal to 0.5 rather than instantaneous fluctuations.

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What is a continuous wave field and how is its energy measured?

A field assigns a definite physical quantity to every location in space and time. A scalar field assigns a single number, such as temperature or particle density, while a vector field assigns a magnitude and direction, such as electric field strength E or magnetic field B.

Harmonic waves propagate oscillating field values through space according to the phase factor kx - omega t, where the wavenumber k = 2 pi / lambda relates to wavelength lambda and angular frequency omega = 2 pi nu relates to cyclic frequency nu. The wave crests travel at the phase speed c = lambda nu.

Because optical frequencies oscillate hundreds of trillions of times per second (for example, 600 THz corresponds to green-cyan light with wavelength lambda = c / nu = 499.65 nm), measuring instruments record the time-averaged energy flux rather than the instantaneous field oscillation. Over any complete period, the average of cos^2(omega t) is exactly 0.5.

When an isotropic source emits total power P into three dimensions, the energy spreads evenly over concentric spherical wavefronts of area 4 pi r^2. The intensity at distance r is given by I = P / (4 pi r^2). For a 1 W point source, the intensity is 0.0795775 W/m^2 at distance r = 1 m, and drops to one-quarter (0.0198944 W/m^2) at distance r = 2 m.

The Doppler shift in sound is an illustrative analogy for wave frequency shifts: sound waves travel through a material medium (air), which creates an asymmetry between a moving source and a moving listener. This analogy has a strict physical limit: light requires no material ether medium, and in 1905 Einstein demonstrated that electromagnetic wave transformation depends purely on relative velocity between observers.

One worked example

I = \frac{P}{4\pi r^2}, \quad \langle \cos^2(\omega t) \rangle = \frac{1}{T}\int_0^T \cos^2(\omega t)\,dt = 0.5

Intensity equals total power divided by four pi r squared, and the time average of cosine squared over a period is one half.

At frequency nu = 600 THz (6.0e14 Hz) with light speed c = 299792458 m/s, the wavelength is lambda = c / nu = 499.65 nm. A 1 W source produces intensity I = 1 / (4 pi (1)^2) = 0.0795775 W/m^2 at 1 m, and I = 1 / (4 pi (2)^2) = 0.0198944 W/m^2 at 2 m.

A stopping point: A field is a value assigned to every place; observed optical intensity is a time average, not an instantaneous pulse.

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