Waves are all around us. Electromagnetic waves carry the information sent to your phone. Ocean waves can propel your boat to shore. Guitar strings move in a wave-like pattern, which is then translated into sound waves. And the newly discovered gravitational waves are further evidence for Einstein's theory of general relativity. (What was the displacement at LIGO which showed the existence of gravitational waves?)
We can start by classifying all the different types of waves.
Mechanical waves exist in a material medium, such as on a string, in the air, through the earth's crust, or on a cable.
Electromagnetic waves can travel through air or the vacuum of space. They don't need a material medium. They travel at a defined value of 299, 792, 458 m/s.
These are the waves of protons, neutrons, electrons, atoms, and molecules. They are unfamiliar to our everyday experience, however scientists working to build the next level of technology are familiar with them. The wavelength of one of these matter waves is \( \lambda = \frac{h}{p} \), where \(h\) is Planck's constant and \(mv\) is the momentum.
These were first proposed by Oliver Heaviside in the late 19th century, and then further developed by Henri Poincaré and Einstein. Huge cosmic events cause gravitational waves, such as colliding black holes, colliding neutron stars, supernovae, and the like. Read more about them here.
\begin{equation} v = \lambda f \qquad (\textnormal{Periodic wave}) \tag{1} \end{equation} \begin{equation} k = \frac{2\pi}{\lambda} \end{equation}
\begin{equation} y(x,t) = A \, \textnormal{cos} \left( \omega \, (\frac{x}{v} - t) \right) = A \, \textnormal{cos} \left( 2 \pi f \, (\frac{x}{v} - t) \right) \qquad (\textnormal{Wave moving in + x}) \tag{2} \end{equation} \begin{equation} y(x,t) = A \, \textnormal{cos} \left( 2 \pi \, (\frac{x}{\lambda} - \frac{t}{T}) \right) \end{equation} \begin{equation} y(x,t) = A \, \textnormal{cos} (k x - \omega t) \end{equation} \begin{equation} \frac{\partial^2 \, y(x,t)}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 \, y(x,t)}{\partial t^2} \qquad (\textnormal{Wave equation}) \tag{3} \end{equation} \begin{equation} v = \sqrt{\frac{F}{\mu}} \qquad \textnormal{(waves on a string)} \end{equation}
\begin{equation} P_{avg} = \frac{1}{2} \omega^2 A^2 \sqrt{\mu F} \qquad \textnormal{(Average power, sinusoidal wave)} \end{equation} \begin{equation} \frac{I_1}{I_2} = \frac{r_2^2}{r_1^2} \qquad \textnormal{(Inverse-square law for intensity of wave)} \end{equation}
\begin{equation} \xi \, (x, t) = \xi_1 \, (x, t) + \xi_2 \, (x, t) \qquad \textnormal{(Wave superposition)} \end{equation}
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