To view any of the movies, click on the blue underlined text in the description.
Convergence of Fourier Series
SQUARE WAVE - This movie shows a sequence of partial sums for the Fourier series of a square wave. The function is discontinuous, so the convergence is like 1/n, and the Gibbs overshoot occurs at the discontinuities.
SAWTOOTH WAVE - The function is f(x) = x on [-1,1]. Although the function is continuous on (-1,1), the periodically extended function has discontinuities at the endpoints. Thus the convergence is slow (like 1/n) and the Gibbs overshoot is visible.
QUADRATIC FUNCTION - The function is f(x) = x^2 on [-1,1]. In this case the periodically extended function is continuous, but has a discontinuous derivative at the endpoints. The convergence is fairly rapid (like 1/n^2) and there is no Gibbs overshoot. Note the "ringing" at x = 0, and the somewhat slower convergence at x = 1 and -1 where the derivative is discontinuous.
Fourier Sine and Cosine Series
FOURIER SINE SERIES - The function expanded is f(x) = x + x^2 in a Fourier sine series on [0,1]. The movie shows the first 51 partial sums. The graphs show the periodic extension of the odd extension of the original function. The Gibbs overshoot is evident near the discontinuities of the extended function.
FOURIER COSINE SERIES - The function expanded is f(x) = x + x^2 in a Fourier cosine series on [0,1]. The movie shows the first 20 partial sums. The graphs show the periodic extension of the even extension of the original function. The extended function is continuous so there is no Gibbs overshoot and the convergence is fairly rapid.
Example of Eigenfunction Expansion
EXAMPLE OF STURM-LIOUVILLE EIGENFUNCTIONS - In this movie, the function f(x) = x(1 - x) is expanded in the eigenfunctions of the following Sturm-Liouville system: y" + ly = 0 on 0 < x < 1, with y(0) = 0 and y'(1) = 0. The movie shows the first 20 partial sums of the series. The struggle to converge is evident at x = 1, where the given function has a non-zero slope and where all of the eigenfunctions have zero slope.
Expansions in Legendre Polynomials
CONVERGENCE OF LEGENDRE EXPANSIONS- These two movies show sequences of partial sums of expansions in Legendre polynomials on the interval [-1,1]. The first movie shows partial sums of polynomials polynomials up to order 51 for a discontinuous function equal to -1 in the left-half interval and 1 in the right-half interval. The graphs show the Gibbs phenomenon near the discontinuity, and also show a considerable struggle to converge at the endpoints. The second movie shows partial sums for a function which is continuous in[-1,1], with a slope discontinuity at 0. The function is f(x) = 0 for [-1,0] and equal to x for (0,1]. The convergence is quite good, with good graphical representation of the function with partial sums including only polynomials of degree 10 and less.
Expansions in Bessel Functions
CONVERGENCE OF FOURIER-BESSEL EXPANSIONS- These two movies show sequences of partial sums of expansions in Bessel functions on the interval [0,3]. The first movie shows partial sums of the expansion of the function f(r) = 1 in a series of J0's. The graphs show the Gibbs phenomenon near the endpoint r = 3. There is also considerable oscillation near r = 0, but of a somewhat different nature. The second movie shows partial sums of the expansion of f(r) = 1 in a series of J1's. Again the Gibbs phenonmenon is apparent at the endpoint r = 3. There are also oscillations in the convergence near r = 0, but of a more complicated character.
Diffusion - Cooling of a Slab
COOLING OF A SLAB 1 - This movie shows a time sequence for the cooling of a slab. The slab (which is aluminum and 0.1 m in thickness) has a uniform initial temperture of 100 C. The ends of the slab at = 0 and L are held at zero temperature. The red curve in each graph is the initial temperature. The blue solid curve is the "exact" solution calculated from the first 10 terms of the series solution. The blue dashed curve is the first term only of the series, and is seen to be a good approximation for larger times. The diffusion time for this slab is L^2/(D pi^2) = 14.5 seconds, where D is the thermal diffusivity of aluminum.
COOLING OF A SLAB 2 - This movie shows the same aluminum slab, but now the boundary conditions are zero flux conditions at both ends. The initial temperature varies linearly from 0 at the left end of the slab to 200 C at the right end of the slab. The red curve in each graph is the initial condition. The blue solid curve is the "exact" solution based on the first 10 terms of the series, and the blue dashed curve is a long-time approximation equal to the sum of the constant term and the first non-constant term of the series.
Diffusion - Irreversibility
IRREVERSIBILITY - These two movies illustrate the irreversibility of solutions of the diffusion equation, as shown by loss of information about the initial state as time progresses. In the first movie, the time evolution is shown of three solutions in a finite slab, corresponding to different initial conditions, with all solutions satisfying zero boundary conditions at the edges of the slab. The initial conditions were chosen so that the three solutions all have the same coefficient of the first harmonic at the initial time. As time progresses, the solutions, which are very different initially, coalesce first and then ultimately decay to zero. In the second movie, two solutions are shown. One has an initial condition equal to the first harmonic, and the second has that initial condition plus a noise term proportional to the 20th harmonic, with an amplitude of 10% of the first harmonic. As the movie shows, the noise terms disappear very rapidly, during which time the first harmonic term hardly changes at all.
Diffusion - Newton's Law of Cooling
A MORE COMPLICATED BOUNDARY CONDITION - This movie shows the time evolution of an initial constant temperature in a slab, when the left boundary is insulated and the condition on the right boundary is Newton's law of cooling. The parameter values are such that the fundmental mode has an e-folding time of 31.7 hours. The movie runs from 0 to 50 hours. The temperature curve on the left boundary remains horizontal, as required by the no-flux condition. The initial temperature is a constant 60 C and the ambient temperature to which the right face is exposed is 10 C.
Diffusion - Kelvin's Estimate of Earth's Age
COOLING OF THE EARTH - This movie shows the temperature versus depth in the earth, assuming an intial temperature of 2000 C and a surface temperature maintained at 0 C. The movie shows the time evolution from 0 to 30 million years in increments of 0.5 million years. The fixed red line has a slope equal to the present day value of the temperature gradient at the surface. The moving blue line shows the temperature gradient at the surface at each time. When the blue line and the red line coincide, the elapsed time is the time since the cooling process started -- i.e., the age of the earth. In this model, that age turns out to be about 25 million years. We now know the true age is much greater. The model underestimates the age because of a physical process that Kelvin could not have known about -- the heat liberated by the decay of radioactive elements in the earth's crust.
Laplace Equation - Contours
CONTOUR PLOT FOR CHANGING BOUNDARY CONDITIONS - The Laplace equation describes equilibrium. Any change in the boundary conditions leads to a change in the entire interior solution. This movie illustrates such a dependence. The movie shows the effects of a continuous change in the boundary condition, using contour plots. At the beginning of the movie, the potential is zero on the bottom and sides of the rectangle, and is a sine function of unit amplitude on the upper boundary. As the movie progresses, the amplitude of the sine on the upper boundary is gradually diminished, and a sine of gradually increasing amplitude is imposed on the right boundary. At the end of the sequence, the potential on the top is zero, and the potential on the right side is a sine of unit amplitude.
Laplace Equation - Far Field Approximation
FAR FIELD AND PENETRATION OF BOUNDARY NOISE. These two movies illustrate certain features of solutions of the Laplace equation in the two-dimensional semi-infinite geometry given by 0 < x < 1 and y > 0. In both cases, the boundary potential is zero on the two sides x = 0 and x = 1, and is zero as y goes to infinity. Both movies show plots of the solution versus x as y is continuously increased -- that is, as we rise above the boundary at y = 0. In the first movie, we see three different solutions corresponding to three different boundary conditions, chosen so that all three solutions have the same lowest order far-field approximation. As y increases, the three solutions coalesce and then ultimately decay to zero. This shows that different boundary functions can appear the same if we are far enough from the boundary. In the second movie, we compare two solutions as we rise above the boundary. The first solution is a single low order harmonic on the boundary (the n = 1 harmonic), and the second solution on the boundary is the same low order harmonic plus some high frequency noise (n = 20). As y increases, we see the high frequency component rapidly disappears, illustrating the very limited penetration of high order harmonics into the interior.
Wave Equation - Guitar Modes
GUITAR MODES. These three movies show how the response of a guitar string depends on the point at which the string is plucked. The first movie shows dynamically the first six modes of the string when the string is excited by plucking at the midpoint. The second movie shows the first six modes when the excitation is very near the bridge of the guitar, at 0.95 of the length of the string. The third movie shows a bar graph of the relative energy in each of the first 10 modes, as the point of plucking is changed continuously from the midpoint to 0.995 of the total length.
Standing Waves in a Coffee Cup
COFFEE SLOSH. These five movies show standing wave modes in a cylindrical coffee cup of radius 4 cm. The height of the coffee in the cup is 8 cm. In all movies except the last, the nodal surfaces are shown in white. The first movie shows the lowest radially symmetric mode. The second movie shows the second radially symmetric mode. The third movie shows the sloshing mode -- the lowest radial mode for the first angular harmonic. This is the mode responsible for spills when you walk with a full cup of coffee. The fourth movie shows the second radial mode for the first angular harmonic. The last movie, an example of a higher mode which is not so easy to excite, is the fifth radial mode for the fourth angular harmonic.