MyOpenMath/Solutions/Standing wave trick

A quick way to find the wavelength of the n-th mode is to draw a single standing wave and allow the length, L, to be the variable. In this figure L is an integral multiple of a half wavelengths:
 * $$n\frac \lambda 2 = L_n$$

This equation is true regardless of whether L or &lambda; is the parameter that varies. So we we solve for &lambda and make the following replacements:
 * $$L_n \rightarrow L \text{ and } \lambda \rightarrow \lambda_n.$$

This yields,
 * $$\lambda_n = \frac{2L}{n}.$$

Off-subject commentary
Is this "trick" worth showing physics students? The answer is "yes" and "no". We begin with the conventional answer:

No
The only students who might need this insight are physics students planning to get a post graduate degree in physics. Tricks like this make it easier to score well on the prelims or GRE for such a degree.

Yes
Despite the soundness of the previous answer's logic, there are two reasons for posting this trick on Wikiversity:
 * 1) The fundamental purpose for taking an introductory physics course is not to introduce equations, but to establish that equations may be obtained by sound reasoning. Any student could benefit from one unifying lesson associated with this "trick": It illustrates the fact that one can sometimes the simplest way to analyze something is to deliberately NOT draw the diagram to scale.
 * 2) Elsewhere I have discussed the use of undergraduate prelims to reduce the cost of higher education.  Over the years, my attempts to construct a pass-fail prelim for introductory physics has taught me that this is a non-trivial task.  Students need to be told what to study, and in order to keep the costs down, students need free access to practice questions.  And, if the study materials for this prelim are to be posted, we also need to post exactly what equations will be given (otherwise preparation for the prelim becomes an exercise in the memorization of equations.)
 * One more pair of features is required for this prelim to be successful: While individual instructors and institutions won't be tasked with designing the prelims, we need to give these instructors and institutions the flexibility to exert control over the design of a prelim.  Will students be given the equation for the standing waves of a violin string?  Or, will the students be expected to derive it from first principles?  For a subset of all institutions that would offer an undergraduate prelim, a study-guide that shows students how to quickly "derive" this standing-wave formula might be the best option.
 * The most challenging part of designing a free and open studyguide for these prelims is the fact that there can be very few "secret" questions because such secrets are difficult to keep. They are also expensive because individual institutions will each want to create their own collections of these surprise questions.  Fortunately the application of these prelims as a pass-fail filter to allow students to take a lower-cost truncated version of a college course will permit the use of a lower quality test based on publicly available studyguides.