# Standing Waves - The Actual Bit *For the prerequisite knowledge feel free to check out* [[Interference Basics]]! When two waves moving in opposite directions but with the same amplitudes and frequencies meet, they create standing waves. In short, these waves don't move - they seemingly oscillate A standing wave is defined strictly with the equation: $y'(x,t) = 2y_{m}\sin{(kx)}\cos {\omega t}$ >[!Success]- Deriving the equation >Since a standing wave is the addition of two like transverse waves which travel in the opposite direction, the waves can be written as: >$y_{1}(x,t) = y_{m}\sin(kx-\omega t)$ >$y_{2}(x,t) = y_{m}\sin(kx+\omega t)$ >Adding this requires a [[Trig Identities - Derivations DLC Ver. (Maths)#Double Angle Identities|double angle identity.]] Try it out, you should get: >$y'(x,t) = [2y_{m}\cos(kx)]\sin(\omega t)$ >Which is the general formula for finding standing waves. What's actually happening to a standing wave is that wave pulses are getting sent and being reflected so fast that our eyes (more our brain) can't process what's happening, leading to a seemingly stationary ## Nodes and Antinodes In standing waves, the wave doesn't move at all - it's been destructively interfered with - in certain spots. These are called *nodes* and form the basis of *harmonics*. ![[Pasted image 20241026202544.png]] Strictly speaking, nodes occur when the [[Interference Basics#Superposition Waves Ver. - Interference Basics|phase difference]] of the wave allows for fully destructive interference, while antinodes where the wave is fully constructive! Let's double back to our [[Waves Foundations#More Transverse Waves|transverse waves]] for a second. We're going to want to find a phase that gives us $sin \ kx = 0$, so let's write that down: $kx = n\pi$ For an antinode, we can just use the equation: $kx = \left( n+\frac{1}{2} \right)\pi$ >[!Success]- Why? >In our case (and with the wave equation) the amplitude of the resulting standing wave will be $2ym_{m}\sin(kx)$. For a node to occur, this has to be 0 as the wave is fully destructively interfered with. For an antinode to occur, this has to be at its maximum, so $sin (kx)$ must equal to 1. ## Harmonics, Overtones! Standing waves have both *harmonic* and *overtone* modes, mostly meaning that the standing wave oscillates much faster than it should. In all fairness, both these terms basically mean the same thing - the *first overtone* mode is the same as the *second harmonic* mode. One way we typically visualise this is through organ pipes. As a result, we've got three different types of 'pipe' we can create a standing wave in: ![[Pasted image 20240410014448.png]] Yeah, it's terrible. Also, to get the length of the tube, we can use the formula: $\lambda = \frac{2L}{n}$ Mind you, this doesn't hold for the tube with 1 open and closed end. Since even numbered harmonics give us fully destructive interference (prove this!), we can't actually # A Standing Wave Case Study: The Rayleigh-Jeans Law Thermodynamics is hard. Everything becomes a probability when you get to a certain level, so all you need up doing is solving and integrating under probabilistic [[Distributions (Maths)|distributions]]. Back in the late 1800s, there was an issue. We'd gotten data of the sun's spectra, we'd been able to plot a frequency-against-wavelength graph of the data, but we had no model to really explain the shape of the model. %%and find that. NOW!!!%% For further reading, check out my (primary) source: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Mechanics/02._Fundamental_Concepts_of_Quantum_Mechanics/Deriving_the_Rayleigh-Jeans_Radiation_Law. Find fun concepts to tackle here! # Resonance # Example Questions # What's Next? And that's that for the basics of waves! It's time to press forward - time to specialise into EM or Acoustics! - [[Intro to Electromagnetic Waves (Personal)]] - Start your EM Journey here, if you choose to go down the path of an optician! - [[All about Acoustics (Sound Waves)]] - Venture into sound through this open door! Not much at the moment but looking to add more soon.