Suppose you repeatedly dip your finger into a sink full of water to make circular waves. What happens to the wavelength of these waves as you move your finger faster?
The wavelength increases
The wavelength decreases
The wavelength remains the same
This is one of the first justification questions I ask after spring break in general physics, the night after I've introduced waves and v = lf. At this point in the year, everyone knows to use either an equation or calculation to justify an answer appropriately.
Even with such a straightforward equation, my guys will often wrap their minds in a huge knot attempting this kind of justification. Why? Well, some folks think that moving the finger "faster" means the wave speed increases. But even for those who start on the right track, they get confused because they don't recognize what quantity doesn't change. Their reasoning will sound something like:
"The faster finger means that the frequency increases. If f goes up then v goes up, by v = lf with v in the numerator. And since the velocity increases, wavelength must also increase, because l is in the numerator of the equation v = lf."
The trick to justifying an answer qualitatively with equations is to start with the UNCHANGING quantity. I train the class to start each of these kinds of questions with asking themselves, Based on facts of physics, what must stay constant?
Then the correct justification leaps out at them: "Okay, the speed of a wave depends on the material (fact of physics). Since the wave is still traveling through the same water, the speed v doesn't change. Now, we want to find wavelength, so rearrange to l = v/f. The numerator doesn't change. Frequency increases, because the finger produces more waves per second. With f increasing in the denominator, wavelength must decrease."
Not only does identifying the unchanging quantity avoid confusion, it also guarantees some partial credit. Even if this student had said frequency decreased, he would have earned something for stating that speed doesn't change, and attempting to use this fact in his justification.