AP Labs

1) Waves: There are few ideas that are getting confused here. In the traveling wave (slinky) lab you created the wave at whatever frequency you chose to shake the end of the slinky. This frequency was imposed by you and had nothing to do with the properties of the spring. The wave speed, c, was determined by the mass per unit length and the tension and that's all[ except for friction with the floor which tends to slow things a bit, especially for waves that are weak in movement any way]. So frequency and wavespeed are independent.

Do not confuse this externally imposed frequency with the resonant frequency of an oscillator given by 2 pi omega. The frequency in the wave lab was imposed by you, not by the materials.

For the questions referring to two differing media ( like that Latin , huh?) note that except for the floor friction, which was substantial for the coil ( heavy spring), the tension in the two media had to be equal. You can't pull harder on one end then the other unless the springs were accelerating down the hallway. If they were, I missed it.

2) Standing waves. A standing wave is a different situation. This is where a single pulse, or small imposed oscillation of a frequency which is related to the wavespeed, causes large oscillations because waves reflected from both ends reinforce each other. In this case, say for two fixed ends L meters apart, the harmonic frequency IS related to wavespeed by f = c/2L or 2 c/2L or 3 c/2L etc. [This is because the wave must satisfy the condition that wavelength = 2 L/integer so that both ends are nodes and nodes are an integer number of half wavelengths apart. This is the same as requiring that reflections from the two ends cause the right kind of interference at nodes ( destructive) and antinodes (constructive).] Since in this case f = integer c/2L, a higher wavespeed and higher frequency are associated. If you try to impose a frequency other than a harmonic you will not get a standing wave pattern.