Section 1.4: Exploring the Input of Data: Formulas
Please wait for the animation to completely load.
In many animations you will be expected to enter a formula to control the animation (position is given in centimeters and time is given in seconds). Restart. In this Exploration, you are to enter a classical wave function, y(x,t), for a wave on a string into the text box. This initializes the wave function at t = 0 on the graph. Once you have done this, the time evolution of the wave function is governed by the form of the wave function and the "resume" and "pause" buttons.
There are a few important rules for entering functions. Notice that the default value in the text box, exp(-(x-t)*(x-t)), corresponds to the wave function e^{−(x−t)(x−t)}. Note that the product in the argument of the exponential is (x-t)*(x-t) and NOT (x-t)(x-t). This is the way the computer understands multiplication. You must enter the multiplication sign, *, every time you mean to multiply two things together. Remove the * and see what happens. You get an error and you can see what you entered. Division is represented as x/2 and NOT x\2. In addition, the Physlet understands the following functions:
sin(a) | cos(a) | tan(a) | sinh(a) | cosh(a) | tanh(a) | |
asin(a) | acos(a) | atan(a) | asinh (a) | acosh(a) | atanh(a) | |
step(a) | sqrt(a) | sqr(a) | exp(a) | ln(a) | log(a) | |
abs(a) | ceil(a) | floor(a) | round(a) | sign(a) | int(a) | frac(a) |
where "a" represents the variable expected in the function (here it is x, t, or a function of the two).
Try the following real functions for y(x,t) and describe what each change does to the time-dependent wave function:
- exp(-(x-2*t)*(x-2*t))
- exp(-(x+2*t)*(x+2*t))
- exp(-(x-4*t)*(x-4*t))*cos(2*x)
- sin(2*pi*x-2*pi*t)
- sin(pi*x)*cos(pi*t)
Try some other functions for practice.
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