P4: Reflection and Transmission

In this question, we will knot two pieces of rope together, with the same tension tau but different masses per unit length Mu1 and Mu2. The knot where the density step occurs is at the middle, X12=L/2=5. Select StepDown and StepUp from the preset menu, and watch what happens. As the ``incident'' pulse hits the knot, part is reflected and part is transmitted: sometimes the pulse flips over! We'll label quantities to do with the incident pulse with a ``I'', reflected with ``R'', and transmitted with ``T''; stuff having to do with the first (blue) part of the string we label with 1 and the second (red) part with 2. We use ``Reflectionless'' boundary conditions, terminating the string so the pulses won't keep bouncing around and confuse us (how we do that is another story...) Let's investigate this as quantitatively as we can without stealing the excitement from the problem sets.

(P4a) Step Down in Density

Load the preset StepDown. Notice that we've chosen a narrower incident peak for this problem. Read off the FWHM width of the incident peak in time,

, and the amplitude

from the Configure menu; while you're doing so, copy down the values for Mu1 and Mu2 for this preset.

Either by measurement, or by calculation from the FWHM in time and the speed of sound V1 on the first segment of string, find the FWHM width of the peak in space. Include it in your writeup.

Measure the amplitude and the FWHM width of the reflected pulse. Measure the amplitude and the FWHM width of the transmitted pulse. Include these in your writeup.

It's easy to calculate quantitatively the widths of the different pulses! Calculate, using your measured widths for the pulses and using the speed of sound in the two media, the widths of the transmitted and reflected pulses in time, and . They will agree. Include in your writeup.

Why are they all the same? Think of the knot (the black rectangle) as exerting a force on the red string as the incident pulse hits it. Surely the force lasts just as long as the incident pulse lasts: hence the transmitted pulse must last the same time (and have the same overall shape) as the incident pulse! Similarly, the knot, pulled by the incident pulse, can reflect energy back only so long as the incident pulse is there: again, the times must agree.

Let's get a qualitative feeling for the amplitudes of the reflected and transmitted pulses. Suppose the red string were extremely light: it then would seem much the same as a free boundary condition (how can wiggling a spider web thread put any significant force on a rope?) The reflected pulse should thus look the same as that for a free boundary condition. Does that agree with what you see? The incident and reflected pulse off the spider thread will have the same height =. When the incident pulse is centered just on the knot, the knot on the blue rope is thus at Y = + = 2 . That means that the transmitted pulse will have twice the amplitude of the incident pulse! Our Mu2 is one quarter of Mu1 (not really a thread); is the amplitude of the transmitted pulse bigger than the incident? In your writeup, explain how energy can be conserved, when the transmitted pulse is both higher and longer!

There are two equations that we will use in the homework to derive a formula for the transmitted and reflected amplitudes for a step jump in the mass density per unit length. Both involve the fact that stuff from the red string and the blue string have to agree at the knot. Since the peak of the transmitted and reflected pulses form at the knot exactly when the incident pulse peaks there, and since the string can't jump there (or it would break), we get an equation guaranteeing the continuity of string:

.
In your writeup, try your measured values for

, and

in this equation. To how many decimal places does your measurement agree with it?

The second equation reflects the conservation of energy at the knot between the two strings. Since the knot cannot absorb or emit energy, the power flowing into the knot must be zero at all times. The power coming into the knot from the blue rope is + (remember is negative, so adding it to makes smaller reflecting the power leaving the knot in the reflected pulse). The power leaving the knot on the red string is . So we get a conservation of energy equation, yielding the balance of power:

Changing the top plot (versus X) to ``Power'', measure the peak power in the incident pulse. (There are two equal pulses: pick either one.) Run past the knot, and measure the peak powers of the transmitted and reflected pulses. (You might wonder that the integrated power looks much larger on the red string than it did in the original pulse. The integral of P(x,t)dx isn't particularly meaningful: it's the integral of P(x,t)dt that measures the energy flow, and the red string pulse is moving faster...) Verify the conservation-of-energy equation: to how many decimal places does it appear valid? Write the percentage error in your lab write up.

(P4b) Step Up in Density

Load the preset StepUp: here the mass per unit length Mu2 is larger than Mu1. Copy these densities into your writeup. Measure the amplitudes, widths in time

, widths in space

, and peak powers P for the incident, transmitted and reflected waves. Include into your writeup.

Make sure the 's are the same, as before. Check that the continuity-of-string equation is satisfied. Include both into your writeup.

Explain qualitatively the amplitude of the reflected pulse, by thinking about the limit where the second rope becomes infinitely massive. Figure out the kind of boundary condition you would expect the blue string to see at the knot in this limit, and include in your writeup. For this boundary condition, what is the sign and amplitude of the reflected pulse?

In the infinitely massive limit, what should be the amplitude of the transmitted pulse, according to the continuity-of-string equation above? Predict this in your writeup.

Verify that your StepUp measurements of the amplitudes satisfy the continuity-of-string equation, and that the measurements of the powers satisfy the conservation-of-energy equation: add to your writeup how many decimal places are satisfied for each equation according to your measurements.

Keep the measurements of the amplitudes, powers, time widths, and spatial widths for these two different mass densities: they will likely appear on problem set in the near future.

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