REVIEW: Waves on a String
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1 Lecture 14: Solution to the Wave Equation (Chapter 6) and Random Walks (Chapter 7) 1 Description of Wave Motion REVIEW: Waves on a String We are all familiar with the motion of a transverse wave pulse on a taut string, or a slinky toy. This is motion in two dimensions (x, y) where the x direction is along the string. The y direction is transverse to the string and measures the amplitude of the wave. The motion is also time dependent. Thus the solution is of the form y(x, t). Transverse waves on a string have a differential equation of motion as follows: 2 y t = 2 y 2 c2 x 2 where c is a speed parameter. For a string under tension T and having a mass per unit length value µ the speed is given by c = T/µ. At first sight, this equation seems to resemble the LaPlace or Poisson equations which we solved using the relaxation method. However, that method is not applicable to the wave equation because we are seeking a time-dependent, not a stationary solution. However, as with the LaPlace equation, we will consider the solution y(x, t) to be a set of grid points y(i, n). Similarly, we will impose a boundary condition by keeping the ends of the string fixed. Finally, we will need, as in any motion problem, and initial (t = 0) condition describing the string. Numerical Solution to the Wave Equation Just as in the LaPlace equation, we can write a finite difference version of the wave equation which will lead to an integration equation for our numerical solution. In terms of the grid points approximation y(i, n), this differential equation looks like y(i, n + 1) + y(i, n 1) 2y(i, n) ( t) 2 c 2 We can then write the iteration solution as [ ] y(i + 1, n) + y(i 1, n) 2y(i, n) ( t) 2 y(i, n + 1) = 2 [ 1 r 2] y(i, n) y(i, n 1) + r 2 [y(i + 1, n) + y(i 1, n)] r c t x In the iteration on the left side we have the new time value t = (n + 1) t, while on the right side we have the two previous time values t = n t and t = (n 1) t. In order to start the iteration we must specified the condition of the string at all points for two time intervals. The usual approach is to say that the string has a specific say Gaussian pulse form for the two time intervals before the calculation starts. y 0 (x) = exp [ k(x x 0 ) 2] The x 0 parameter gives the center of the Gaussian pulse while the k parameter gives the width of the pulse. As stated before, we need to impose boundary conditions on the string at the two ends. The simplest first choice is to have the ends fixed: y(0, n) = y(m, n) for all times n, and where the length of the string L = M x.
2 Lecture 14: Solution to the Wave Equation (Chapter 6) and Random Walks (Chapter 7) 2 REVIEW: Numerical Solution to Waves on a String Example 6.1 in Figure 6.2 The pseudo-code for solving the wave equation is given in Example 6.1 on page 159, with the results shown in Figure 6.2. A C/C++ implementation of that pseudo-code is the fixedstring.cpp program which we will study in this class. The most complicated part of this program is producing the cascade of snapshots of the string at different times, which can eventually be animated. We will explore the different possibilities of the r parameter as discussed on page 161. Study of Numerical Instabilities When doing numerical solutions to non-analytic problems it is always important to check for numerical instabilities as well as for convergence. The wave equation solution shows one type of numerical instability which can be inferred from physics. Normally one thinks that by decreasing the step size then the solution will become more inaccurate. As it happens, this is not true with the wave equation solution. The iteration equation from the previous page is y(i, n + 1) = 2 [ 1 r 2] y(i, n) y(i, n 1) + r 2 [y(i + 1, n) + y(i 1, n)] r c t x From the above you can see that there are two step sizes being used: x and t. The iteration equation makes use of the ratio of those two steps in the r ratio factor. The fixedstring.cpp program has a default value r = 1, which can be changed by a command line option fixedstring [rratio] input. Effectively the r ratio parameter links the spatial step size x with the time step size t. In the program the spatial step x is taken as 0.01 m, while the wave speed c is taken as 5.0 m/s. Hence, the time step t is computed as /5.0 = seconds. fixedstring [rratio] So as a first exercise, run the fixedstring program without specifying an rratio input. Then run the program with just a slight increase of rratio, say 1.1. This would mean that time step is being increased to seconds. You will find that the program fails with some obscure error. To see what has really happened, open the text output file fixedstring.dat which is actually being used by the plotting function. This text file has three columns giving the time (seconds), position along the string (x, meters), and the amplitude (y, meters). You can see that the amplitude becomes wildly divergent. The reason for this instability is that the with a time step of seconds, the wave will have moved cm. However, this is more than the step size cm. The wave is moving faster than our algorithm can notice. In other words x/ t < c. You can also try input values of rratio < 1. For those case you will not see a divergence in the wave amplitude. However, there will still be inaccuracies. You can notice these by monitoring the amplitude of the wave pulses. As we have seen the wave splits into two equal pulses traveling in opposite directions. It must be that the amplitudes of these two pulses always add up to the original unit amplitude when the two pulses overlap completely in space.
3 Lecture 14: Solution to the Wave Equation (Chapter 6) and Random Walks (Chapter 7) 3 Numerical Solution to Waves on a String Example Figure 6.3 for a Composite String You know from first year physics when a wave is traveling along a composite string of two different mass/length values then there will be a partial reflection and transmission of the wave at the boundary. The different segments of the string have different wave speeds. This wave behavior is analogous to how light rays behave when they go from air into glass. There is partial transmission for the refracted light ray in the glass, and also a partial reflection back into the air. As an in class exercise, copy the fixedstring.cpp file to another version fixedstringsplit.cpp. Then modify this program to use a second input argument, call it vratio. This will indicate the ratio of the speeds of two segments of the string. For simplicity, take the split in the string to be at the halfway mark, 0.50 m. You will then have to introduce a modified r factor into the iteration equation which is implemented in the propagatewave function. It should take you about 30 lines of additional code to make such a modified program. See if you can reproduce the wave behavior as shown in Figure 6.3 on page 163. Note that you would always have to take vratio value of less than 1.0 because of the numerical instability problem discussed in the previous page. Homework Assignment Do exercises 6.3, 6.4, and 6.5. Please produce postscript, gif, or other types of figures formats which are readable on any computer.
4 Lecture 14: Solution to the Wave Equation (Chapter 6) and Random Walks (Chapter 7) 4 Physics of Random Systems Random Systems So far all of our physics problems have been completely deterministic. A given set of initial conditions and/or boundary conditions is prescribed, a potential or force function is specified, and then a differential equation is solved numerically. In principle for classical (non-quantum) physics all problems are deterministic. However, when large ensembles of particles are being studied, as in molecular physics when Avogadro s number of particles is being considered, then it is a practical impossibility to study the motion of individual particles. Instead, a statistical approach is taken. One studies the average behavior of a large group of particles. In such cases of statistical physics the microscopic two-body force or potential is replaced by a phenomenological approach. Such is the case for diffusion of one set of molecules into another, for example the case of a drop of cream spreading into a cup of coffee. The black coffee will eventually turn brown via the cream molecules dissipating more or less uniformly into the liquid. The physics of such diffusion processes are studied with a Random Walk approach. The Random Walk in One Dimension Consider the following simple example in one dimension x. A person is in the center of a room. The person can take one step of exactly one foot long in either the forward or the backward direction. The direction forward or backward is determined by a flip of a coin: heads means the person goes forward one step, while tails means the person goes backward one step. The coin is flipped every unit of time. This is called a Random Walk because the direction of any given step is completely random. In a sense this simple example is like the motion of the cream molecules in coffee. Their movement is controlled by the random interactions with the more numerous coffee molecule. The main difference is that the cream-in-coffee case is in three dimensions, and the step sizes are not fixed. The first question to be asked is how far does the person go in say 20 flips of the coin? After you have answered that question, the next example is to consider say 500 such individuals doing the same coin-flip exercise. On average, how far do those 500 people move after each has flipped the coin 20 times? The fixedwalk1d program The fixedwalk1d.cpp program implements the random walk process. We will study the outputs of this program in class. On the following page, we have a mathematical analysis of the random walk process.
5 Lecture 14: Solution to the Wave Equation (Chapter 6) and Random Walks (Chapter 7) 5 Mathematical Analysis for Random Walk in One Dimension Analyzing the mean square displacement From the computer program, or from Figure 7.2 on page 185, we see that the squared displacement averaged over the total number of walkers grows linearly with time according to the diffusion Equation 7.1 < x 2 >= 2Dt The coefficient D is called the diffusion coefficient, and t is the total time which is linearly proportional to the number of steps taken. The factor of 2 is used by convention in this field of physics. The average of the x 2 is taken because obviously the average of x itself will be zero for all the walkers. For each walker we can compute the total displacement after n step as x n = s i where s i is the step size, either +1 or 1. From this result we can compute the square x 2 n easily enough x 2 n n = s i s j = s i s j j=1 j=1 Now the s i s j will be randomly +1 or 1, unless i = j for which s i s i = 1. Hence x 2 n = s 2 i = n since the total time t is proportional to the number of steps n, we get the diffusion equation and D = 0.5 is the theoretical prediction. x 2 n = 2Dt One can carry out a similar statistical analysis (page 186) to get the fluctuation in < x 2 > from its average value < (x 2 n < x 2 n >) 2 =< x 4 n > < x 2 n >= 2n 2 2n For a large number of walkers the fluctuation in the displacement < x 2 > from its average goes as 2n 2.
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