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What should it be, [itex]{sin(ωt-\frac{k}{n}x)}[/itex] or [itex]{sin(ωt-nkx)}[/itex]? I am contemplating this with respect to the proper waveform in a medium, following specific rules, particularly when it comes to the dispersion relation and ultimately what this would mean in the context of the Lorentz transformations.
A single wave is described by the expression [itex]{sin(ωt-kx)}[/itex], where [itex]{ω}[/itex] is the angular frequency and [itex]{k}[/itex] is the wavenumber. In transparent media the refractive index should be included into the waveform. Also the speed of the wave is given by the ratio of whatever multiplies [itex]{t}[/itex] to whatever multiplies [itex]{x}[/itex]. Taken this as a rule of thumb, the waveform becomes
regards,
Alfred.
A single wave is described by the expression [itex]{sin(ωt-kx)}[/itex], where [itex]{ω}[/itex] is the angular frequency and [itex]{k}[/itex] is the wavenumber. In transparent media the refractive index should be included into the waveform. Also the speed of the wave is given by the ratio of whatever multiplies [itex]{t}[/itex] to whatever multiplies [itex]{x}[/itex]. Taken this as a rule of thumb, the waveform becomes
[itex]{sin(ωt-\frac{k}{n}x)}[/itex] (1)
where [itex]{n}[/itex] can be any real value. Applying the rule of thumb, the propagation velocity, [itex]{u}[/itex], is[itex]{u=\frac{ωn}{k}}=c[/itex] (2)
In textbooks however, the waveform with a refractive index is given by [itex]{sin(ωt-nkx)}[/itex] and applying the rule of thumb leads to [itex]{\acute{u}=\frac{ω}{nk}}[/itex], which is different from Eq. (2). This form is based on the optical path length (OPL), which is defined as the product of the geometric length of the path that light follows through the medium and the index of refraction, in other words the physical length of the path is multiplied by the refractive index. To make sure [itex]{\acute{u}=\frac{ω}{nk}}[/itex], [itex]{\frac{ω}{k}}[/itex] is equated with [itex]{c}[/itex] and it leads to [itex]{\acute{u}=\frac{c}{n}}[/itex], conforming to Eq. (2). However, Eq. (1) suggests a virtual path length through the medium, as opposed to OPL. The virtual path length can be defined by a transformation where [itex]{\acute{t}=t}[/itex] and [itex]{\acute{x}=\frac{x}{n}}[/itex]. Accordingly,[itex]{Δ\acute{x}=\frac{Δx}{n}}[/itex] (3),
which represents a length contraction. It is the effect of reduced depth conception when observing objects in an aquarium filled with water. The objects look closer (and also larger in size) than they physically are, and vice versa, observing objects in air from water, they will look further away, as can be experienced and deduced by rearranging Eq. (3). Eq. (3) is useful to locate the position of virtual objects. It also suggests that length contraction is an optical illusion.regards,
Alfred.