At this point we have some idea about traveling waves both conceptually and mathematically. Now we move into the first of the settings where we explore the consequences of that understanding. The particular setting we address first is standing waves. The key features of the setting the provoke us to think about the possibility of standing waves are that there are two waves traveling in the same medium in opposite directions. These waves have the same frequency and roughly the same amplitude (ideally exactly the same amplitude). In addition there are boundary conditions that define the behavior of the wave are specific points in the physical setting -- usually at the ends but sometimes in other places as well.
First -- the first clip is a quick review of the standard solution of the wave equation and a discussion of the different ways that symbols are defined. It is worth taking the time to notice that the definition used are all consistent with our language just with a different emphasis in Walt's class at MIT.
Wave Review: (3 min)
Then we will move on to a discussion of how having two waves moving in opposite directions leads to a wave where the spatial and temporal parts are separated instead of mixed. Walt moves really quickly through this assuming that the trig identities that he is using are obvious to his listener. Take the time to remind yourself of the trig identities for the sums and different of angles and the half angle formulae. After that take some time and make sure you understand why this odd looking function he writes down means that what happens in the x direction is independent of what is happening in time. Be sure it makes sense to you what a node is and why.
Standing Wave Math: (4 min)
The next step is actually a very important one conceptually that we will explore from a different perspective in class. This clip is a traditional way to approach some understanding of the meaning and importance of boundary conditions (and initial conditions) in both physics and math.
Boundary Conditions: (4 min)
Ultimately what you need to be able to do is use your understanding of the plots of standing waves to describe and predict what will happen in various settings. Walt introduces the relevant plots and the language of normal modes, fundamentals, resonance, overtones and harmonics. It is a confusing collection of terms which all seek to describe the same set of features in slightly different ways depending on whether you're thinking like a musician, a physicist, or a chemist.Try to sort out all those different descriptors and a clear sense of the meaning of nodes and antinodes. It seems likely that this will take several trips through the clip.
Plots of Standing Waves solutions: (7 min)
Finally -- it's always helpful to see standing waves in the flesh as it were and Walt does a nice version of the standard demo for this concept. I will do the same demo if I can remember to bring my giant spring/string to class.
Standing Wave Demo: (5 min)
These last two clips are important to go through because of the conceptual challenges associated with sound waves relative to transverse waves. We won't get to the concepts in these clips until later in the week so you can explore these ones later. Be careful......there are some very conceptually challenging distinctions between what is happening to the physical movement of the air molecules in a sound wave and what is happening to the pressure as an alternative way of descibing the sound wave. The demo that Walt does is a classic physics lab exercise for this course which he does it more convincingly than we would be likely to do it in our lab. The equipment that he is using is called Kundt's Tube.
Kundt's Tube Math: (6 min)
Kundt's Tube Demo: (6 min)