>[!summary] There are two types of waves: > Transverse waves are waves that travel perp to the direction of oscillation. Longitudinal waves are waves that travel parallel to direction of oscillation. > The speed of a wave is: $v = \lambda f$ > A standing wave is a type of wave where there are nodes at the start and end of the full length of the wave. > In general: $\lambda _n = \frac{2L}{n}$ > Where the frequency: >$\begin{array}{c} f_n = n\frac{v}{2L} \quad Open \\ f_n = n\frac{v}{4L} \quad Closed \end{array}$ > If there are more than one wave we can determine where and how the wave will interfere with each other. > >$\begin{array}{c} r_1 - r_2 = n\lambda \quad Constructive\\ r_1 -r_2 = (n+1/2)\lambda \quad Destuctive \end{array}$ # Types of Waves Transerve waves are when a wave travels perp to the direction of oscillation. We know the figure below is travelling as a transverse wave at point P because the direction P moves in with respect to the direction is perp. ![[wav_1.png]] [^1] >[!note] Explanation This describes a traverse wave over time. The direction of the displacement at any point is always perpendicular point P Longitudinal Wave are a type of wave that travels in the direction of motion. We know the below figure is travelling as a longitudinal wave because the direction motion is in the same direction as the direction of propagation ![[wav_2.png]] [^1] >[!note] Explanation This describes a longitudinal wave overtime. # Derivation of the wave expression For a wave to move it must be given energy and hence a velocity. And our wavelengths can be found from the peak-peak or crest-crest amount. ![[wav_3.png]] [^1] >[!note] Explanation A wave overtimme marking amplitude, and wavelength So a general solution to this is: $v = \lambda f$ where v is our wavespeed and f is our freq Note that our wavefunction of a equation has both and x and t direction. So we need to develop a new equation for both direction. From [[Simple Harmonic Motion]] if assume this as a SHM string our wave equation is $y(t) = Acos(\omega t + \phi) $ Notice that our function as time following a cos graph, but is depdent on the $\lambda$ over a distance $2\pi$ so: $y(x) = Acos(kx + \phi) $ where k is $\frac{2\pi}{k}$ # Standing Waves Standing waves are a type of closed waves. They are defined as having nodes at the start and end point. If we assume a box to form our standing waves, the standing waves must create a node at the end of the boxes. Many differnt types of waves can fit this pattern hence a general equation: $\lambda _n = \frac{2L}{n}$ If we assume $v = \lambda f$ than solviong for f we get: $f_n = \frac{n}{2L}\sqrt{\frac{F}{\mu}}$ Note this is true for a closed wall on both sides. If we assume the wave is open on both ends or closed on one end, our frequecny change. $\begin{array}{c} f_n = n\frac{v}{2L} \quad Open \\ f_n = n\frac{v}{4L} \quad Closed \end{array}$ ![[wav_4.png]] ![[wav_5.png]] [^2] >[!note] Explanation Overview of the different types of wavelength and frequency for different harmonics. # Constructive & Destructive Interference Constructive interference refers to when two wave function interference with each other so there waves add up to each other. Destructive interference refers tow hen the waves add up to cancel each other out. $\begin{array}{c} r_1 - r_2 = n\lambda \quad Constructive\\ r_1 -r_2 = (n+1/2)\lambda \quad Destuctive \end{array}$ [^1]: Taken from https://tikz.net/waves/ by Izaak Neutelings (December 2020) [^2]: Taken from https://tikz.net/waves_standing/ by Izaak Neutelings (December 2020) --- > 📚 Like this note? [Star the GitHub repo](https://github.com/rajeevphysics/Obsidan-MathMatter) to support the project and help others discover it! ---