I understand resonance for a simple harmonic oscillator but not for more complex systems like standing waves.
How can I be in resonance with the normal mode in an organ pipe?
I understand that the frequency of the force acting on the system has to match the natural frequency of the column of air in the pipe. However, the force acting on the system is generating pulses of pressure waves onto the system (suppose I'm blowing air or something) and I just don't see how nodes are going to be preserved if I'm continuously sending these pulses! Namely, I'm imagining that the pulses acting on the system will disturb the nodes of whatever harmonic was present in the organ pipe. The same with a string.
Also, what exactly is a normal mode? My textbook says that standing waves can only satisfy this equation:
Longitude of rope = (lambda/2)N or anything similar that depends on the system.
The thing is I'm so darn sure I saw another type of standing wave in my physics lab, where there was a half-wavelength on one extreme of the rope that was much shorter than all the rest of half-wavelengths. In fact, I think it wasn't even a half-wavelength it was a quarter-wavelength!
Also, I keep reading that a guitar string normally vibrates according to the fundamental frequency. I've seen countless videos that demonstrate in slow-motion how the vibrating string has hundreds of crests and is clearly not in its fundamental frequency. Other sources say harmonics coexist at the same time, this makes little sense to me right now.
Finally, is the topic of waves something I will understand more clearly later on in my studies as a physics major? I've heard you study this topic a big-deal in Differential Equations. Is this true? I've only seen CalcI and CalcII.