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Sunday, February 24, 2008

References for Fast Fourier Transform and Matlab

The Fourier Transformation and its Application QA403.5 B7
The Fast Fourier Transformation and its application QA403 B75 1988

-------------------------

Online:
Wave function with fft and ifft in matlab from Physics Forum
Fourier Transform Example 2 Matlab code (a copy below)

-----------------------
Fourier Transform Example 2 Matlab code

%      ***** MATLAB Code Starts Here *****
%

%FOURIER_TRANSFORM_02_MAT

fig_size = [232 84 774 624];

m2ft = 3.2808; % conversion from meters to feet

g = 32.2; % gravitational acceleraton (ft/s^2)

beta = 180; % relative wave direction (deg)

z = 0; % depth below the surface (ft)

rho = 1.999; % water density (lbm/ft^3)

w = logspace(-1,1,401); % vector of frequencies with log spacing

k = w(2)/w(1); % ratio of adjacent frequencies

delta_w = [diff(w) (k-1)*w(401)]; % differences between adjacent frequencies

w = w + delta_w .* rand(1,length(w)); % random selection of frequencies

H_13 = [0.1 0.5 1.25 4] * m2ft; % wave heights to be tested (ft)

a = 0.0081; % PM spectrum coefficients

b = 0.74;

for i = 1:length(H_13)

Upm = 12.4 * sqrt(H_13(i));

w0 = g / Upm;

Sa = (a*g^2 ./ w.^5) .* exp(-b*(w0./w).^4); % PM spectrum (frequency component)

Sb = (2/pi) * cos(beta*pi/180)^2; % PM spectrum (directional component)

Sc = rho * g * exp(-w.^2 * z / g); % PM spectrum (depth component)

S(i,:) = Sa .* Sb .* Sc; % total PM spectrum

end

%Parameters for the notch filter

num1 = [1 0 0.64]; den1 = [1 1.6 0.64]; % first stage numerator and denominator

num2 = [1 0 4]; den2 = [1 4 4]; % second stage numerator and denominator

num = conv(num1,num2); % total filter numerator and denominator

den = conv(den1, den2);

[mag,ph] = bode(num,den,w); % magnitude and phase of the notch filter

mag = mag'; ph = ph';

%Calculation of filter outputs in the frequency domain

for i = 1:length(H_13)

out(i,:) = mag .* S(i,:); % filter output for each sea state input

end

%Calculation of the time-domain wave signal for sea state 2

%and the output of that signal from the notch filter

t = [0: 0.125: 800]; % time vector

phi = 2*pi*(rand(1,length(w)) - 0.5); % random phase of ith frequency

A = sqrt(2*S(2,:).*delta_w); % amplitude of ith frequency

for i = 1:length(t)

wave(i) = sum(A .* cos(w*t(i) + phi));

wave_out(i) = sum(mag .* A .* cos(w*t(i) + phi + ph));

end

%Sample Fourier Transforms

fft_wave = fft(wave); % Fourier transform of wave signal

fft_wave_out = fft(wave_out); % Fourier transform of notch filter output

ws2 = pi / 0.125; % one-half of the sampling frequency

ww = linspace(0,ws2,3200); % frequency vector for sample Fourier transforms

[i1,i2] = min(abs(ww - 5)); % selecting the relevant frequencies

%Plot the data

figure(1),clf,subplot(211),semilogx(w,mag),grid,xlabel('Frequency (r/s)'),...

ylabel('Magnitude'),title('Notch Filter Magnitude Function'),z=axis;axis([0.1 10 z(3) z(4)]),...

subplot(212),semilogx(w,ph),grid,xlabel('Frequency (r/s)'),...

ylabel('Phase (deg)'),title('Notch Filter Phase Shift'),z=axis;axis([0.1 10 z(3) z(4)]),...

set(gcf,'Position',fig_size)

figure(2),clf,subplot(221),semilogx(w,S(1,:),w,out(1,:)),grid,xlabel('Frequency (r/s)'),...

ylabel('Magnitude'),title('Sea State 1 Input & Output'),z=axis;axis([0.1 10 z(3) z(4)]),...

subplot(222),semilogx(w,S(2,:),w,out(2,:)),grid,xlabel('Frequency (r/s)'),...

ylabel('Magnitude'),title('Sea State 2 Input & Output'),z=axis;axis([0.1 10 z(3) z(4)]),...

subplot(223),semilogx(w,S(3,:),w,out(3,:)),grid,xlabel('Frequency (r/s)'),...

ylabel('Magnitude'),title('Sea State 3 Input & Output'),z=axis;axis([0.1 10 z(3) z(4)]),...

subplot(224),semilogx(w,S(4,:),w,out(4,:)),grid,xlabel('Frequency (r/s)'),...

ylabel('Magnitude'),title('Sea State 4 Input & Output'),z=axis;axis([0.1 10 z(3) z(4)]),...

set(gcf,'Position',fig_size)

figure(3),clf,subplot(211),plot(t,wave),grid,xlabel('Time (s)'),ylabel('Height (ft)'),...

title('Wave Height, Sea State 2'),z = axis;,...

subplot(212),plot(t,wave_out),grid,xlabel('Time (s)'),ylabel('Height (ft)'),...

title('Measured Wave Height at Output of Notch Filter, Sea State 2'),axis(z),...

set(gcf,'Position',fig_size)

figure(4),clf,subplot(211),plot(ww(1:i2),abs(fft_wave(1:i2)),'o'),grid,xlabel('Frequency (r/s)'),...

ylabel('Magnitude'),title('Sample Fourier Transform of Wave Signal'),z = axis;,...

subplot(212),plot(ww(1:i2),abs(fft_wave_out(1:i2)),'o'),grid,xlabel('Frequency (r/s)'),...

ylabel('Magnitude'),title('Sample Fourier Transform of Notch Filter Output'),axis(z),...

set(gcf,'Position',fig_size)

%
%      ***** MATLAB Code Stops Here *****

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References for Methods of Applied Math

Methods of Applied Math 2 from wisc

Linear and Nonlinear waves , G.B. Whitham, Wiley 1974. [Research Monograph, a classic]
Elementary Applied Partial Differential Equations, Richard Haberman, THIRD Edition, Prentice-Hall 1998. [Undergrad-Grad Textbook, Separation, Green's functions background, Chap 12-14: graduate applied Math]
Partial Differential Equations of Applied Mathematics, Erich Zauderer, 2nd Ed., Wiley-Interscience 1989. [Graduate, Textbook]
Similarity, Self-Similarity and Intermediate Asymptotics, G.I. Barenblatt, Cambridge [Research Monograph]

Less applied references (some would say `purer')

Partial Differential Equations , F. John, Springer-Verlag, 4th edition, 1986. [Chaps 1, 2]
Partial Differential Equations , L.C. Evans, American Mathematical Society 1999. [Chaps 1-4, Textbook for Math 819, not applied]
Partial Differential Equations , R. McOwen, Prentice-Hall 1996. [Not applied math, but nice concise overview]

Friday, February 8, 2008

Phase v.s. Group velocities

This post is a combination of the following websites, without the applets, tho. To see the applet please visit the author's website.

A demonstration of group velocity
Group velocity demonstration
Phase and group velocities
Phase, group, and Signal velocities

A demonstration of group velocity:

Group Velocity and Phase Velocity

The phase velocity is fixed at 1.0 (arbitrary units).

About

Understanding the Pattern

To find out what is happening here, first set the group velocity equal to one using the slider bar. You will now see a frozen pattern moving steadily to the right on the screen. This represents the amplitude of, say, a sound wave propagating to the right. Now increase the frequency using the lower slider bar. You will see the peaks crowding closer together. But why are they coming in these bunches? The applet is actually plotting the wave generated by two pure notes which are very close together, so beats are generated, a waa waa waa sound, each waa corresponding to one bunch of peaks, or one wavepacket as we shall call it.

As long as the group velocity is set equal to one, the wavepackets move at the same speed as the individual waves. This is true for ordinary sound and light waves. But other kinds of waves, such as surface water waves, and quantum electron waves, have more interesting behavior. The bow wave of a moving ship, viewed where it has fanned out some distance from the ship, looks like a single wavepacket, which typically includes a few individual wave peaks. If you look closely, you will see that the wave peaks move relative to the wavepacket, in fact they go at twice the speed! In our applet, the speed of the individual waves, called the phase velocity, is always set equal to one. The group velocity is the speed of the wavepacket. So, to get a picture of how the individual waves behave in the bow wave wavepacket from a ship, set the group velocity equal to 0.5. You will see individual waves disappearing at the leading edge of the wavepacket. (Of course, in our applet they move into the next packet, but the bow wavepacket is a single packet, which is constructed by beating many close notes together rather than just two.)

The wavepacket describing a nonrelativistic electron in quantum mechanics is also like a "single beat" but in this case the group velocity is twice the phase velocity, so as the wave propagates the individual waves disappear at the trailing edge of the packet.

A complete discussion of the quantum case can be found in my lecture on Wavepackets.

More Physics Applets!

Group velocity demonstration

Demonstration of Group Velocity

The difference between group velocity and phase velocity can be quite difficult to grasp, but at the most basic descriptive level the difference is quite simple. The quantity we commonly refer to as the "speed-of-light" is its phase velocity: the speed at which the wavefronts of a pure frequency tone travel. The group velocity is the speed at which pulses (or "wave packets"), which are made up of a range of spectral components, travel. The group velocity and phase velocity are the same in a vacuum, but in any other medium this is not, in general, the case.

The Java applet in the window below is intended to help you to visualise the concept of group velocity by allowing you to observe what happens when two waves of slightly different frequency propagate in a medium in which the refractive index (or, equivalently, the propagation constant) is a function of frequency.

In the window you see three waves. The red wave has a fixed frequency and propagation constant. (If you are curious, the "refractive index" for this wave is 1.2, and the "free space velocity" is 1% of the frame width per update, so you can see that the units are somewhat arbitrary!) The green wave has a slightly higher frequency, and the refractive index it experiences is controlled by the slider at the bottom. The slider allows you to offset the refractive index of the green wave from the red wave by plus/minus 0.2. The blue wave is obtained by summing the red and green waves, and represents the total propagating field. You can see that the interference pattern formed by the two original waves consists of a rapidly oscillating wave inside a slowly varying envelope. (Communications engineers might prefer to see this as a carrier which has been amplitude-modulated with a low frequency sine wave. What are the carrier and modulation frequencies in this case?)

You can see that when the refractive indices at the red and green wavelengths are equal, the envelope travels at the same speed as both the other waves: the group velocity is equal to the phase velocity. However, if you increase the refractive index at the green wavelength, by moving the slider to the right, the green wave slows down. The interference pattern between the red and green waves is no longer fixed, if you look carefully you can see that the "carrier" wave now moves through the modulation envelope. As a result the envelope has a higher (spatial) frequency, i.e. the pulses are shorter, and they travel more slowly than either of the original waves. The group velocity is now less than the phase velocity. Note that the shortening of the pulses is completely consistent with this: as they travel from a region of higher group velocity (e.g. where n_r = n_g) to a region of lower group velocity (e.g. where n_r < n_g) they must bunch-up to compensate, just as the wavelength of light is shortened when it travels from a region of lower refractive index to higher refractive index.

What happens if you move the slider to the left, i.e. making the refractive index difference negative? Can you make sense of this?

Maintained by:   Mark Summerfield (m.summerfield@ee.unimelb.edu.au)
Created:  11 August 1998
Last Modified:  21 October 1999
Copyright © 1998 The University of Melbourne. Disclaimer

Phase and group velocities

The following movies show wave packets with various combinations of phase and group velocity. Each waveform is the sum of five sinusoids with the general format

which yields [Packet Picture]

Each movie is set up for seamless looping, but you may have to download and play them outside your browser to enable the loop feature.

Phase Velocity = Group Velocity [179K]: A=0.7, w_o = k_o = 1, Dw = Dk = 0.05
The entire waveform—the component waves and their envelope—moves as one. Kind'a boring.

Phase Velocity > Group Velocity [179K]: A=0.7, w_o = k_o = 1, Dw = 0.025, Dk = 0.05
The component waves move more quickly than the envelope.
Phase Velocity <> [179K]: A=0.7, w_o = k_o = 1, Dw = 0.1, Dk = 0.05
The component waves move more slowly than the envelope.
Group Velocity = 0 [18K]: A=0.7, w_o = k_o = 1, Dw = 0, Dk = 0.05
The envelope is stationary while the component waves move underneath.
Phase Velocity = 0 [178K]: A=0.7, w_o = 0, k_o = 1, Dw = Dk = 0.05
Now only the envelope moves over stationary component waves.
Phase Velocity = -Group Velocity [179K]: A=0.7, w_o = k_o = 1, Dw = -0.05, Dk = 0.05
Finally, the envelope moves in the opposite direction of the component waves.

Phase, group, and Signal velocities

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) = A₀cos(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 v_p. In terms of the cyclical frequency and wavelength we have v_p = 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 v_p, 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/v_p where c is the speed of light in vacuum and v_p is the phase velocity of light in that medium. Now, since v_p = 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 k_A = 2 rad/meter and an angular frequency of w_A = 2 rad/sec, so it's individual phase velocity is v_A = 1 meter/sec. The sinusoidal wave denoted as "B" has a wave number of k_B = 2.2 rad/meter and an angular frequency of w_B = 8 rad/sec, so it's individual phase velocity is v_B = 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 mis-construed as ultra-high-speed signals. In fact, by simply setting k_A to 2.2 and k_B 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 w and 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

Another common source of confusion regarding the propagation speed of periodic physical effects is the phenomenon of "numerator dynamics", discussed in the note on Lead-Lag Frequence Response.

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Group velocity

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The group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope of the wave) propagate through space. For example, imagine what happens if you throw a stone into the middle of a very still pond. When the stone hits the surface of the water, a circular pattern of waves appears. It soon turns into a circular ring of waves with a quiescent center. The ever expanding ring of waves is the group, within which one can discern individual wavelets of differing wavelengths traveling at different speeds. The longer waves travel faster than the group as a whole, but they die out as they approach the leading edge. The shorter waves travel slower and they die out as they emerge from the trailing boundary of the group.

The group velocity v_g is defined by the equation

$v_g \ \equiv\ \frac{\partial \omega}{\partial k}\,$

where:

ω is the wave's angular frequency;

k is the wave number.

Note: The above definition of group velocity is only useful for wavepackets, which is a pulse that is localized in both real space and frequency space. Because waves at different frequencies propagate at differing phase velocities in dispersive media, for a large frequency range (a narrow envelope in space) the observed pulse would change shape while traveling, making group velocity an unclear or useless quantity.

The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave. In most cases this is accurate, and the group velocity can be thought of as the signal velocity of the waveform. However, if the wave is travelling through an absorptive medium, this does not always hold. Since the 1980s, various experiments have verified that it is possible for the group velocity of laser light pulses sent through specially prepared materials to significantly exceed the speed of light in vacuum. However, superluminal communication is not possible in this case, since the signal velocity remains less than the speed of light. It is also possible to reduce the group velocity to zero, stopping the pulse, or have negative group velocity, making the pulse appear to propagate backwards. However, in all these cases, photons continue to propagate at the expected speed of light in the medium. ^[1] ^[2] ^[3] ^[4]

The function ω(k), which gives ω as a function of k, is known as the dispersion relation. If ω is directly proportional to k, then the group velocity is exactly equal to the phase velocity. Otherwise, the envelope of the wave will become distorted as it propagates. This "group velocity dispersion" is an important effect in the propagation of signals through optical fibers and in the design of high-power, short-pulse lasers.

Anomalous dispersion happens in areas of rapid spectral variation with respect to the refractive index. Therefore, negative values of the group velocity will occur in these areas. Anomalous dispersion plays a fundamental role in achieving backward propagating and superluminal light. Anomalous dispersion can also be used to produce group and phase speeds that are in different directions [ 2 ]. Materials that exhibit large anomalous dispersion allow the group velocity of the light to exceed c and/or become negative [ 4 ].

The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877^[5].

[hide]

[edit] Matter wave group velocity

Albert Einstein first explained the wave-particle duality of light in 1905. Louis de Broglie hypothesized that any particle should also exhibit such a duality. The velocity of a particle, he concluded then (but may be questioned today, see above), should always equal the group velocity of the corresponding wave. De Broglie deduced that if the duality equations already known for light were the same for any particle, then his hypothesis would hold. This means that

$v_g = \frac{\partial \omega}{\partial k} = \frac{\partial (E/\hbar)}{\partial (p/\hbar)} = \frac{\partial E}{\partial p}$

where

E is the total energy of the particle,

p is its momentum,

$\hbar$ is Dirac's constant.

Using special relativity, we find that

$\begin{align} v_g &= \frac{\partial E}{\partial p} = \frac{\partial}{\partial p} \left( \sqrt{p^2c^2+m^2c^4} \right)\\ &= \frac{pc^2}{\sqrt{p^2c^2 + m^2c^4}}\\ &= \frac{p}{m\sqrt{(p/(mc))^2+1}}\\ &= \frac{p}{m\gamma}\\ &= \frac{mv\gamma}{m\gamma}\\ &= v. \end{align}$

where

$m$ is the rest mass of the particle,

c is the speed of light in a vacuum,

γ

is the Lorentz factor.

and $v$ is the velocity of the particle regardless of wave behavior.

Quantum mechanics has very accurately demonstrated this hypothesis, and the relation has been shown explicitly for particles as large as molecules.

[edit] See also

Dispersion (optics) for a full discussion of wave velocities
Phase velocity
Slow light

[edit] References

^ George M. Gehring, Aaron Schweinsberg, Christopher Barsi, Natalie Kostinski, Robert W. Boyd, “Observation of a Backward Pulse Propagation Through a Medium with a Negative Group Velocity”, Science. 312, 895-897 (2006).
^ Gunnar Dolling, Christian Enkrich, Martin Wegener, Costas M. Soukoulis, Stefan Linden, “Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial”, Science. 312, 892-894 (2006).
^ A. Schweinsberg, N. N. Lepeshkin, M.S. Bigelow, R.W. Boyd, S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber”, Europhysics Letters. 73, 218-224 (2005).
^ Matthew S Bigelow, Nick N Lepeshkin, Heedeuk Shin, Robert W Boyd, “Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities”, Journal of Physics: Condensed Matter. 18, 3117-3126 (2006)
^ Brillouin, Léon. Wave Propagation and Group Velocity. Academic Press Inc., New York (1960)

Tipler, Paul A. and Ralph A. Llewellyn (2003). Modern Physics. 4th ed. New York; W. H. Freeman and Company. ISBN 0-7167-4345-0. 223 p.

[edit] External links

Greg Egan has an excellent Java applet on his web site that illustrates the apparent difference in group velocity from phase velocity.
Group and Phase Velocity - Java applet with configurable group velocity and frequency.
Maarten Ambaum has a webpage with movie demonstrating the importance of group velocity to downstream development of weather systems.

Phase velocity

From Wikipedia

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The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase speed is given in terms of the wave's angular frequency ω and wave vector k by

$v_\mathrm{p} = \frac{\omega}{k}$

In a dispersive medium, the phase velocity varies with frequency and is not necessarily the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will propagate.

The phase velocity of electromagnetic radiation may under certain circumstances (e.g. in the case of anomalous dispersion) exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer. It was theoretically described by physicists such as Arnold Sommerfeld and Leon Brillouin. See dispersion for a full discussion of wave velocities.

[hide]

[edit] Matter wave phase

In quantum mechanics, particles also behave as waves with complex phases. By the de Broglie hypothesis, we see that

$v_\mathrm{p} = \frac{\omega}{k} = \frac{E/\hbar}{p/\hbar} = \frac{E}{p}$ .

Using relativistic relations for energy and momentum, we have

$v_\mathrm{p} = \frac{E}{p} = \frac{\gamma m c^2}{\gamma m v} = \frac{c^2}{v} = \frac{c}{\beta}$

where E is the total energy of the particle (i.e. rest energy plus kinetic energy in kinematic sense), p the momentum, $γ$ the Lorentz factor, c the speed of light, and β the velocity as a fraction of c. The variable v can either be taken to be the velocity of the particle or the group velocity of the corresponding matter wave. See the article on group velocity for more detail. Since the particle velocity $v < c$ for a massive particle according to special relativity, phase velocity of matter waves always exceed c, i.e.

$\text{[math]}$ c \," src="http://upload.wikimedia.org/math/3/6/9/369a580932f2075fb7cd19554332ef48.png">,

and as we can see, it approaches c when the particle velocity is in the relativistic range. The superluminal phase velocity does not violate special relativity, for it doesn't carry any information. See the article on signal velocity for detail.

[edit] External links

Subluminal, a Java applet

Group and Phase Velocity - Java applet showing the difference between group and phase velocity.

[edit] See also

[edit] References

Tipler, Paul A. and Ralph A. Llewellyn (2003). Modern Physics. 4th ed. New York; W. H. Freeman and Company. ISBN 0-7167-4345-0. 222-3 pp.
Leon Brillouin "Wave Propagation And Group Velocity" Academic Press Inc., New York and London (1960) ISBN 0-1213-4968-3.
Main, Iain G. (1988).Vibrations and Waves in Physics. 2nd ed. New York; Cambridge University Press. ISBN 0-5212-7846-5. 214-6 pp.

Dispersive Waves

Link

In a nondispersive wave medium, waves can propagate without deformation. Electromagnetic waves in unbounded free space are nondispersive as well as nondissipative and thus can propagate over astronomical distances. Sound waves in air are also nearly nondispersive even in the ultrasonic frequency range. If not, that is, if high frequency notes (e.g., piccolo) and low frequency notes (e.g., base) propagate at different velocities, they would reach our ears at different times, and music played by an orchestra would not be harmonious. Most waves in material media are dispersive, however, and wave forms originally set up are bound to change in a manner that the wave energy is more spatially spread out or dispersed.

Animation 1 below shows propagation of dispersive wave packet and Animation 2 nondispersive wavepacket. In making the animation, 100 sinusoidal waves are superposed for a dispersion relation,

(The dispersion relation describes the ion acoustic wave in a plasma and also approximately shallow water waves. See, for example, A. Hirose et al., Plasma Physics, Vol. 20, p. 1179 (1978), in which response of the ion acoustic wave to an impulse has been analyzed.) Note that in the dispersive case shown in the top animation, long wavelength components propagate faster than short wavelength components. The envelope of the wave packet propagates at the group velocity. Wave ripples propagate at the phase velocity. In contrast, nondispersive wave packet (Animation 2) described by

maintains the original wave form.

Animation 3 shows superposition of two sinusoidal waves in dispersive case, sin(x - t) + sin(1.2x - 1.1t) (group velocity = half of phase velocity), while the last animation shows the case of nondisperve wave, sin(x - t) + sin(1.2x - 1.2t). Note that in the dispersive case (Animation 3), wave propagation is not simple parallel shift.

1. Dispersive wavepacket

> with(plots):
animate(sum(.07*(exp(-(.1*k-3)^2)+exp(-(0.1*k+3)^2))*cos(.1*k*x-.1*k/sqrt(1+.1*(.1*k)^2)*t),k=1..100),x=-4..20,t=0..30,frames=60,numpoints=200,color=red);

[Maple Plot]

2. Nondispersive wavepacket

> animate(sum(.07*(exp(-(.1*k-3)^2)+exp(-(0.1*k+3)^2))*cos(.1*k*x-.1*k/sqrt(1+.0*(.1*k)^2)*t),k=1..100),x=-4..20,t=0..30,frames=60,numpoints=200,color=red);

[Maple Plot]

3. Dispersive sinusoidal waves

> animate(sin(x-t)+sin(1.2*x-1.1*t),x=0..50,t=0..63,numpoints=150,frames=100,color=red);

[Maple Plot]

4. Nondispersive sinusoidal waves

> animate(sin(x-t)+sin(1.2*x-1.2*t),x=0..50,t=0..63,numpoints=150,frames=100,color=red);

[Maple Plot]

Sunday, February 24, 2008

References for Fast Fourier Transform and Matlab

References for Methods of Applied Math

Friday, February 8, 2008

Phase v.s. Group velocities

Demonstration of Group Velocity

Group velocity

From Wikipedia, the free encyclopedia

Contents

[edit] Matter wave group velocity

[edit] See also

[edit] References

[edit] External links

Phase velocity

From Wikipedia, the free encyclopedia

Contents

[edit] Matter wave phase

[edit] External links

[edit] See also

[edit] References

Dispersive Waves

Thursday, February 7, 2008

Statistic Inteference C1 1.1

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