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February 24, 2014

Questions

My eldest turned 18 and voted in her first Primary election this week. This being Texas, she decided to register as a Republican. Which means that, soon, we will start fielding phone calls from political campaigns. So I drafted a set of questions to ask the earnest campaign workers when they call.

Posted by distler at 9:30 PM | Permalink | Followups (9)

February 22, 2014

Lying

Sometimes, for the sake of pedagogy, it is best to suppress some of the ugly details, in order to give a clear exposition of the idea behind a particular concept one is trying to teach. But clarity isn’t achieved by outright lies. And I always find myself frustrated when our introductory courses descend to the latter.

My colleague, Sonia, is teaching the introductory “Waves” course (Phy 315) which, as you might imagine, is all about solving the equation

(1)0=( 2t 2c 2 2x 2)u(x,t)0 = \left(\frac{\partial^2}{{\partial t}^2} - c^2 \frac{\partial^2}{{\partial x}^2}\right) u(x,t)

This has travelling wave solutions, with dispersion relation

(2)ω(k) 2=c 2k 2{\omega(k)}^2 = c^2 k^2

If you study solutions to (1), on the interval [0,L][0,L], with “free” boundary conditions at the endpoints,

(3)ux| x=0,L=0\left.\frac{\partial u}{\partial x}\right\vert_{x=0,L} = 0

you find standing wave solutions u(x,t)=Acos(kx)cos(ckt) u(x,t) = A \cos(k x)\cos( c k t) where the boundary condition at x=Lx=L imposes

(4)sin(kL)=0orkL=nπ,n=1,2,\sin(k L) = 0\qquad \text{or}\qquad k L = n\pi,\, n=1,2,\dots

The first couple of these “normal modes” look like

(5) Layer 1 x = 0 x=0 x = L x=L Layer 1 x = 0 x=0 x = L x=L n=1 n=2\begin{matrix} \begin{svg} <svg width="240" height="102" xmlns="http://www.w3.org/2000/svg" xmlns:svg="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" xmlns:math="http://www.w3.org/1998/Math/MathML"> <g> <title>Layer 1</title> <path id="svg_36902_1" d="m220,101c-100,0 -100,-100 -200,-100" stroke-width="2" stroke="#ff0000" fill="none"/> <line fill="none" stroke="#000000" stroke-width="2" x1="20" y1="51" x2="220" y2="51" id="svg_36902_2"/> <foreignObject height="20" width="40" font-size="16" id="svg_36902_3" y="51" x="0"> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> <annotation encoding="application/x-tex">x=0</annotation> </semantics> </math> </foreignObject> <foreignObject id="svg_36902_4" height="20" width="40" font-size="16" y="51" x="200"> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>L</mi> </mrow> <annotation encoding="application/x-tex">x=L</annotation> </semantics> </math> </foreignObject> </g> </svg> \end{svg} & \begin{svg} <svg width="240" height="102" xmlns="http://www.w3.org/2000/svg" xmlns:svg="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" xmlns:math="http://www.w3.org/1998/Math/MathML"> <g> <title>Layer 1</title> <line fill="none" stroke="#000000" stroke-width="2" x1="20" y1="51" x2="220" y2="51" id="svg_36903_2"/> <foreignObject height="20" width="40" font-size="16" id="svg_36903_3" y="51" x="0"> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> <annotation encoding="application/x-tex">x=0</annotation> </semantics> </math> </foreignObject> <foreignObject id="svg_36903_4" height="20" width="40" font-size="16" y="51" x="200"> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>L</mi> </mrow> <annotation encoding="application/x-tex">x=L</annotation> </semantics> </math> </foreignObject> <path fill="none" stroke="#ff0000" stroke-width="2" d="m20,1c50,0 50,100 100,100c43,0 50,-100 100,-100" id="svg_36903_5" fill-opacity="0"/> </g> </svg> \end{svg}\ n=1&n=2 \end{matrix}

To “illustrate” this, in their compulsory lab accompanying the course, the students were given the task of measuring the normal modes of a thin metal bar, with free boundary conditions at each end, sinusoidally driven by an electromagnet (of adjustable frequency).

Unfortunately, this “illustration” is a complete lie. The transverse oscillations of the metal bar are governed by an equation which is not even approximately like (1); the dispersion relation looks nothing like (2); “free boundary conditions” look nothing like (3) and therefore it should not surprise you that the normal modes look nothing like (4).

Unfortunately, so inured are they to this sort of thing, that only one (out of 120!) students noticed that something was amiss in their experiment. “Hey,” he emailed Sonia, “Why is the n=1n=1 mode absent?”

Posted by distler at 3:55 PM | Permalink | Post a Comment

February 11, 2014

Naturalness Versus the Weak Gravity Conjecture

Clifford Cheung and his student have a cute paper on the arXiv. The boldest version of what they’re suggesting is that, perhaps, quantum gravity solves the hierarchy problem.

That’s way too glib a summary, but the detailed version is still pretty surprising.

Posted by distler at 8:32 PM | Permalink | Post a Comment

February 7, 2014

Audiophilia

Humans are hard-wired to find patterns.

Even when there are none.

Explaining those patterns (at least, the ones which are real) is what science is all about. But, even there, lie pitfalls. Have you really controlled for all of the variable which might have led to the result?

Posted by distler at 12:27 PM | Permalink | Followups (8)