P3: Boundary Conditions and Colliding Pulses

(P3a) Fixed Boundary Conditions and Pulse-Antipulse Annihilation

Start again from the default values. (If you're not sure, you can either restart the simulation, or select Default from the ``presets'' menu.) Notice that the right-hand boundary has Fixed boundary condition: at all times y(L)=0. Watch the pulse carefully as it approaches the right-hand boundary: come close, and then set ``Time to Run'' to smaller and smaller values, clicking until you inch in to the wall. If you overshoot and the pulse starts growing again, just type in a negative ``Time to Run'' (the simulation runs backwards too). Notice how the displacement completely vanishes when the center of the packet hits the boundary at L.

Where did the packet go? Observe Y DOT, and explain briefly in your lab writeup.

Now, collide two packets of opposite signs from opposite directions. You can do this by changing the forcing on the right boundary from None to NegPulse, but we'd like you instead to select the preset NegColli (which also splits the string nicely into two colors). Notice that the black square in the middle is motionless. Explain in your lab write-up why this follows from the principle of superposition. (Hint: if the pulse moving right is y(x,t), what is the formula for the pulse moving to the left? What is the sum?

Notice that the motion of the blue curve for the colliding packets in the vicinity of the black rectangle is precisely the same as it was when hitting the right wall with fixed boundary conditions. The wave doesn't care why the point is stationary! This is an extremely useful method for predicting what happens to waves as they approach fixed boundaries: you construct a fictitious (fake) canceling wave from past the end of the string, and let them freely pass through one another at the boundary.

Test your intuition! Suppose I start with an asymmetric pulse. Select the preset ``Asym'', and stop it (be alert!) before it hits the far wall. Write in your lab notebook your guess (no points off for being wrong: commit yourself!) about

Whether the packet will be swallowed by the wall, like the other packet was.
Whether the big bump will point up or down after it hits the wall. (This should be easy.)
Whether the little bump will be ahead (more negative X) or behind (more positive X) after it collides with the fixed boundary at X=L.

After guessing, run the simulation and check your answers. You'll have to look carefully to test for complete swallowing! If you were wrong about any of your guesses, explain the right answer in your writeup.

(P3b) Colliding Pulses Aren't Traveling Waves

Reloading the NegColli preset, run until the two packets are merged together at the black dot. Do this carefully, eventually taking tiny steps in Time to Run. Or, use your estimate of the time for the packet centers to hit the boundary that you did in the introduction to the simulation, plus your knowledge of the wave propagation speed v, to calculate the right amount of time to run from t=0.

As in the previous exercise, before looking with the program, predict in your writeup

What will the potential energy density be for the string when the pulse and negative anti-pulse are completely overlapping?
What will the total energy density be, in terms of the total energies for the original pulses?
What will the kinetic energy density be, in terms of the total energy for the two overlapping pulses?

Check your answers by selecting the three energy densities versus X. Look also at the power: does this make sense? (Hint: energy currents toward the right cancel energy currents toward the left.) If the graph spills out of the window, the easiest way to get the whole graph is to use "Copy Graph" to put it in a new window, and then click the right button (or Control-click the single or left button) in the new graph window: that should ``unzoom'' and show the whole curve. Again, if you guess wrong, explain the right answer in your writeup.

What's true for traveling waves (KE(x)=PE(x), Power(x) = V * u(x)) isn't always true for colliding pulses, standing waves, or anything else. Keep the different formulas straight!

(P3c) Free Boundary Conditions

Reload the ``Default'' preset, and change one or both of the boundary conditions from Fixed to Free. How does this change the motion when the packet hits the boundary?

Watch the wave carefully at the free boundary: zoom in to a small region from X=9 to X=10, leaving the Y-axis to range from -A to 2A. Notice that the slope of the line dy/dx stays zero at the boundary.

In your lab write-up, explain why a force-free boundary must have no slope dy/dx there.

Launch the preset ``Collide'', which takes two packets of the same sign and bashes them together. Run them until they collide: as before, collide them carefully (the point of perfect collision is more obvious when observing Y DOT). Briefly note in your lab write-up (some of these may be obvious)

How the merged packet compares in amplitude to the two individual packets.
The kinetic energy at the point of collision.
The potential energy compared to the total energy at the point of collision.
The derivative dy/dx of the rope at the black dot, as the pulses approach and recede from one another,
The blue rope shape as a function of time, compared to the pulse hitting a free boundary condition.

Again, one can figure out what happens at a free boundary condition by using a fictitious wave coming from past the end of the string - this time, inverted in X but not in Y.

Links Back

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).