Frequency dispersion in groups of gravity waves on the surface of deep water. The red dot moves with the phase velocity, and the green dots propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red dot overtakes two green dots when moving from the left to the right of the figure.
The velocity of a wave can be defined in many different ways, partly because there many different kinds of waves, and partly because we can focus on different aspects or components of any given wave. The ambiguity in the definition of "wave velocity" often leads to confusion, and we frequently read stories about experiments purporting to demonstrate "superluminal" propagation of electromagnetic waves (for example). Invariably, after looking into the details of these experiments, we find the claims of "superluminal communication" are simply due to a failure to recognize the differences between phase, group, and signal velocities.
In the simple case of a pure traveling sinusoidal wave we can imagine a "rigid" profile being physically moved in the positive x direction with speed v as illustrated below.
Clearly the wave function depends on both time and position. At any fixed instant of time, the function varies sinusoidally along the x axis, whereas at any fixed location on the x axis the function varies sinusoidally with time. One complete cycle of the wave can be associated with an "angular" displacement of 2p radians. The angular frequency w of a wave is the number of radians per unit time at a fixed position, whereas the wave number k is the number of radians per unit distance at a fixed time. (If we prefer to speak in terms of cycles rather than radians, we can use the wavelength l = 2p /k and the frequency n = w/2p .) In terms of these parameters we can express a pure traveling wave as the function
A(t,x) = A0 cos(kx – wt)
where the "amplitude" A0 is the maximum of the function. (We use the cosine function rather than the sine merely for convenience, the difference being only a matter of phase.) The minus sign denotes the fact that if we hold t constant and increase x we are moving "to the right" along the function, whereas if we focus on a fixed spatial location and allow time to increase, we are effectively moving "to the left" along the function (or rather, it is moving to the right and we are stationary). Reversing the sign gives A0 cos(kx + wt), which is the equation of a wave propagating in the negative x direction. Note that the function A(t,x) is the fundamental solution of the (one-dimensional) "wave equation"
Since w is the number of radians of the wave that pass a given location per unit time, and 1/k is the spatial length of the wave per radian, it follows that w/k = v is the speed at which the shape of the wave is moving, i.e., the speed at which any fixed phase of the cycle is displaced. Consequently this is called the phase velocity of the wave, denoted by vp. In terms of the cyclical frequency and wavelength we have vp = ln.
If we imagine the wave profile as a solid rigid entity sliding to the right, then obviously the phase velocity is the ordinary speed with which the actual physical parts are moving. However, we could also imagine the quantity "A" as the position along a transverse space axis, and a sequence of tiny massive particles along the x axis, each oscillating vertically in accord with A0 cos(kx -wt). In this case the wave pattern propagates to the right with phase velocity vp, just as before, and yet no material particle has any lateral motion at all. This illustrates that the phase of a traveling wave form may or may not correspond to a particular physical entity. It’s entirely possible for a wave to "precess" through a sequence of material entities, none of which is moving in the direction of the wave. In a sense this is similar to the phenomenon of aliasing in signal processing. What we perceive as a coherent wave may in fact be simply a sequence of causally disjoint processes (like the individual spring-mass systems) that happen to be aligned spatially and temporally, either by chance or design, so that their combined behavior exhibits a wavelike pattern, even though there is no actual propagation of energy or information along the sequence.
Since a general wave (or wavelike phenomenon) need not embody the causal flow of any physical effects, there is obviously there is no upper limit on the possible phase velocity of a wave. However, even for a "genuine" physical wave, i.e., a chain of sequentially dependent events, the phase velocity does not necessarily correspond to the speed at which energy or information is propagating. This is partly a semantical issue, because in order to actually convey information, a signal cannot be a simple periodic wave, so we must consider non-periodic signals, making the notion of "phase" somewhat ambiguous. If the wave profile never exactly repeats itself, then arguably the "period" of the signal must be the entire signal. On this basis we might say that the velocity of the signal is unambiguously equal to the "phase velocity", but in this context the phase velocity could only be defined as the speed of the leading (or trailing) edge of the overall signal.
In practice and common usage, though, we tend to define the "phase" of a signal with respect to the intervals between consecutive local maxima (or minima, or zero crossings). To illustrate, consider a signal consisting of two superimposed sine waves with slightly different frequencies and wavelengths, i.e., a signal with the amplitude function
As most people know from experience, the combination of two slightly unequal tones produces a "beat", resulting from the tones cycling in and out of phase with each other. Using a well-known trigonometric identity we can express the two components of this signal in the form
Therefore, adding the two terms of A(x,t) together, the products of sines cancel out, and we can express the overall signal as
This can be somewhat loosely interpreted as a simple sinusoidal wave with the angular velocity w, the wave number k, and the modulated amplitude 2cos(Dkx – Dwt). In other words, the amplitude of the wave is itself a wave, and the phase velocity of this modulation wave is v =Dw/Dk. A typical plot of such a signal is shown below for the case w = 6 rad/sec, k = 6 rad/meter, Dw = 0.1 rad/sec, Dk = 0.3 rad/meter.
The "phase velocity" of the internal oscillations is w/k = 1 meter/sec, whereas the amplitude envelope wave (indicated by the dotted lines) has a phase velocity of Dw/Dk = 0.33 meter/sec. As a result, if we were riding along with the envelope, we would observe the internal oscillations moving forward from one group to the next.
The propagation of information or energy in a wave always occurs as a change in the wave. The most obvious example is changing the wave from being absent to being present, which propagates at the speed of the leading edge of a wave train. More generally, some modulation of the frequency and/or amplitude of a wave is required in order to convey information, and it is this modulation that represents the signal content. Hence the actual speed of content in the situation described above is Dw/Dk. This is the phase velocity of the amplitude wave, but since each amplitude wave contains a group of internal waves, this speed is usually called the group velocity.
Physical waves of a given type in a given medium generally exhibit a characteristic group velocity as well as a characteristic phase velocity. This is because within a given medium there is a fixed relationship between the wave number and the frequency of waves. For example, in a transparent optical medium the refractive index n is defined as the ratio c/vp where c is the speed of light in vacuum and vp is the phase velocity of light in that medium. Now, since vp = w/k, we have w = kc/n. Bearing in mind that the refractive index is typically a function of the frequency (resulting in the "dispersion" of colors seen in prisms, rainbows, etc.), we can take the derivative of w as follows
Hence any modulation of an electromagnetic wave in this medium will propagate at the group velocity
In a medium whose refractive index is constant, independent of frequency (such as the vacuum), we have dn/dk = 0 and therefore the group velocity equals the phase velocity. On the other hand, in most commonly observable transparent media (such as air, water, glass, etc.) at optical frequencies have a refractive indices that increase slightly as a function of wave number and (therefore) frequency. This is why the high frequency (blue) components of a beam of white light are deflected more than the low frequency (red) components as they pass through a glass prism. It follows that the group velocity of light in such media (called dispersive) is less than the phase velocity.
It is quite possible for the phase velocity of a perfectly monochromatic wave of light, assuming such a thing exists, to exceed the value of c, because it conveys no information. In fact, the concept of a perfectly monochromatic beam of light is similar to the idea of a "free photon", in the sense that neither of them has any physical significance or content, because a photon must be emitted and absorbed, just as a beam of light cannot be infinite in extent and duration, but must always have a beginning and an end, which introduces a range of spectral components to the signal. Any modulation will propagate at the group velocity, which, in dispersive media, is always less than c.
An example of an actual physical application in which we must be careful to distinguish between the phase and the group velocity is the case of electromagnetic waves propagating through a hollow magnetic conductor, often called a waveguide. A waveguide imposes a "cutoff frequency" w0 on any propagating electromagnetic waves based on the geometry of the tube, and will not sustain waves of any lower frequency. This is roughly analogous to how the pipes in a Church organ will sustain only certain resonant patterns. As a result, the dominant wave pattern of a propagating wave with a frequency of w will have a wave number k given by
Since (as we’ve seen) the phase velocity is w/k, this implies that the (dominant) phase velocity in a waveguide with cutoff frequency w0 is
Hence, not only is the phase velocity generally greater than c, it approaches infinity as w approaches the cutoff frequency w0. However, the speed at which information and energy actually propagates down a waveguide is the group velocity, which (as we’ve seen) is given by dw/dk. Taking the derivative of the preceding expression for k with respect to w gives
so the group velocity in a waveguide with cutoff frequency w0 is
which of course is always less than or equal to c.
Unfortunately we frequently read in the newspapers about how someone has succeeded in transmitting a wave with a group velocity exceeding c, and we are asked to regard this as an astounding discovery, overturning the principles of relativity, etc. The problem with these stories is that the group velocity corresponds to the actual signal velocity only under conditions of normal dispersion, or, more generally, under conditions when the group velocity is less than the phase velocity. In other circumstances, the group velocity does not necessarily represent the actual propagation speed of any information or energy. For example, in a regime of anomalous dispersion, which means the refractive index decreases with increasing wave number, the preceding formula shows that what we called the group velocity exceeds what we called the phase velocity. In such circumstances the group velocity no longer represents the speed at which information or energy propagates.
To see why the group velocity need not correspond to the speed of information in a wave, notice that in general, by superimposing simple waves with different frequencies and wavelengths, we can easily produce a waveform with a group velocity that is arbitrarily great, even though the propagation speeds of the constituent waves are all low. A snapshot of such a case is shown below. In this figure the sinusoidal wave denoted as "A" has a wave number of kA = 2 rad/meter and an angular frequency of wA = 2 rad/sec, so it’s individual phase velocity is vA = 1 meter/sec. The sinusoidal wave denoted as "B" has a wave number of kB = 2.2 rad/meter and an angular frequency of wB = 8 rad/sec, so it’s individual phase velocity is vB = 3.63 meters/sec.
The sum of these two signals is denoted as "A+B" and, according to the formulas given above, it follows that this sum can be expressed in the form 2cos(kx-wt)cos(Dkx-Dwt) where k = 5, w = 2.1, Dk = 0.1, and Dw = 3. Consequently, the "envelope wave" represented by the second factor has a phase velocity of 30 meters/sec. Nevertheless, it’s clear that no information can be propagating faster than the phase speeds of the constituent waves A and B. Indeed if we follow the midpoint of a "group" of A+B as it proceeds from left to right, we find that when it reaches the right hand side it consists of the sum of peaks of A and B that entered at the left long before the current "group" had even "appeared". This is just one illustration of how simple interfering phase effects can be miss-construed as ultra-high-speed signals. In fact, by simply setting kA to 2.2 and kB to 2.0, we can cause the "groups" of A+B to precess from right to left, which might mistakenly be construed as a signal propagating backwards in time!
Needless to say, we have not succeeded in arranging for a message to be received before it was sent, nor even in transmitting a message superluminally. Examples of this kind merely illustrate that the "group velocity" does not always represent the speed at which real information (or energy) is moving. This stands to reason, because the two cosine factors of the carrier/modulation waveform are formally identical, so we can’t arbitrarily declare one of them to represent the carrier and the other to represent the modulation. Both are required, so we shouldn’t expect the information speed to be any greater than the lesser of the two phase speeds, nor should it exceed the lesser of the phase speeds of the two components A and B. Furthermore, we already know that the transmission of information via an individual wave such as A will propagate at the speed of an incremental disturbance of A, which depends on how dw and dk are related to each other. In the example above we arbitrarily selected increments of wand k, but our ability to do this in a physical context would depend on a great deal of flexibility in the wave propagation properties of the medium. This is where the ingenuity of the experimenter can be deployed to arrange various exotic substances and fields in such a way as to permit the propagation of waveforms with the desired properties. The important point to keep in mind is that none of these experiments actually achieves meaningful superluminal transfer of information.
Incidentally, since we can contrive to make the "groups" propagate in either direction, it’s not surprising that we can also make them stationary. Two identical waves propagating in opposite directions at the same speed are given by
Superimposing these two waves propagating (with synchronized nodes) in opposite directions yields a standing pure wave