Waves on a string, and cost-benefit analyses for AP physics

An interesting and detailed email from Virginia physics teacher Drew Austen, who attended my Richmond workshop back in June, asked:

...You mention [in my Just the Facts: Waves post] that students should know that the speed of a wave depends on the material through which it travels. This is easy enough for waves on a string, since students should understand both tension and linear density, as in Question #50 on the 2009 AP exam. However, the speed of a sound wave depends on bulk modulus and density, and we don’t cover anything about bulk modulus in AP Physics... Any suggestions?

Bulk modulus is not necessary.  In general, it's simply necessary to know that the wave speed is determined by the properties of the material through which the wave moves, without the details about that dependence.  For example, a student should understand the deep water waves can move at a different speed than shallow water waves; however, the student will NOT be expected to know offhand which move faster, or how much faster.

For years I've considered the precise equation relating wave speed to tension and linear density to be unnecessary.  Most wave questions on the AP exam that I recall don't require recall of the exact form of the equation.  For example, problem 8 on the 1998 exam showed a standing wave on a string, then asked how to change the mass hanging from the string's end to get "more loops" in the string.  But the problem was explicit that a bigger tension meant bigger wave speed -- the tricks were to recognize that more mass meant more tension, that "more loops" meant a smaller wavelength, and that the frequency provided by the tuning fork would be unchanged.  

Drew paid closer attention than I to 2009 multiple choice question 50.  It described a wave moving on a string, then asked "which of the following will cause the wave speed to increase?"  One of the options was to use a string with identical length and tension, but a smaller linear density.  Oops -- knowing in general that changing the string's properties somehow changes the wave speed wasn't good enough.  You had to know, either conceptually or mathematically, that a "lighter" string gives faster waves speed. 

Yet, I'm still not going to teach the equation 

.

Why not?  AP Physics B students have to memorize loads and loads of equations already.  I've finally, after fifteen years of teaching AP, been pointed to a question that required use of this equation.  A student who had not been taught the specific relationship between wave speed, tension, and linear density would still have had a good shot at the correct answer, either using intuition about strings, or by eliminating the two answer choices that were obviously incorrect.  

So I'm making a cost-benefit analysis.  By teaching the equation above in addition to everything I teach already, I've given the students the information necessary to garner 3/5 of a point on one AP test over the past 15 years.  That's the benefit.

What's the cost?  Requiring students to learn yet another equation.  Putting this equation on fundamentals quizzes.  Writing homework and quiz problems to use this equation repeatedly.  (Remember, the benefit of teaching the equation only accrues if students actually remember what you taught.  Just writing the equation on the board once is worse than useless.)  

Is the benefit worth the cost?  I say no.  I show in a demonstration that by tightening a string, I produce a standing wave form that requires the wave speed to have increased.  I also show in a demonstration with a guitar that the light strings produce a higher pitch than the heavy strings, for the same string length -- this requires that light strings produce higher wave speeds.  

I don't repeat these facts again, nor do I make the students "responsible" for that facts on quizzes or homework, nor do I ever write the equation relating speed, tension, and linear density.  [I don't even use the term "linear density, usually, unless someone asks.]  I'm willing to sacrifice 3/5 of a point for less confusion.

Nevertheless, I would bet that my students didn't perform any worse than the national average on this question.  Point is, before you reflexively throw everything on the AP "learning objectives" into the gumbo of your course, consider whether simple matzo ball soup from a box might be easier to cook, and taste nearly as yummy.  Apologies in advance to all residents of Louisiana, but I'm *not* a fan of gumbo.

GCJ