Fundamental Physics/Wave

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Wave is defined as periodic oscillation carries energy travels in space. When talking about wave, we talk about wavelength, speed, angular speed, frequency

Sinusoidal wave

For any sinusoidal wave

Its characteristics are list in the table below

 Wave oscillation equation ${\displaystyle {\frac {d^{2}}{dt^{2}}}f(t)=-\omega f(t)}$ Wave function ${\displaystyle f(t)=ASin\omega t}$ Wavelength ${\displaystyle s=\lambda }$ Wave's speed ${\displaystyle v={\frac {s}{t}}={\frac {\lambda }{t}}}$ Wave's angular speed ${\displaystyle \omega =\lambda f}$ Wave's frequency ${\displaystyle f={\frac {1}{t}}}$

Sinusoidal wave source

AC electrical sinusoidal wave generator

An interaction of 2 electromagnets creates an AC electricity that has amplitude varies sinusoidally

${\displaystyle v(t)=ASin\omega t}$

Series LC

Series LC operates at equilibrium satisfy wave equation

${\displaystyle {\frac {d^{2}}{dt^{2}}}i(t)=-{\frac {1}{T}}i(t)}$

that has root of a sinusoidal wave function

${\displaystyle i(t)=ASin\omega t}$
${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$
${\displaystyle T=LC}$

Electromagnetic sinusoidal wave generator

A coil of N turns operates at equilibrium satisfy wave equation

${\displaystyle \nabla ^{2}E(t)=-\omega E(t)}$
${\displaystyle \nabla ^{2}B(t)=-\omega B(t)}$

that has root of a sinusoidal wave function

${\displaystyle E(t)=ASin\omega t}$
${\displaystyle B(t)=ASin\omega t}$
${\displaystyle \omega ={\sqrt {\frac {1}{T}}}=C=\lambda f}$
${\displaystyle T=\mu \epsilon }$

Summary

 Wave Sinusoidal wave Sinusoidal Plane wave Wave equation ${\displaystyle {\frac {d^{2}}{dt^{2}}}f(t)=-\omega f(t)}$ ${\displaystyle \nabla ^{2}E(t)=-\omega E(t)}$  ${\displaystyle \nabla ^{2}B(t)=-\omega B(t)}$ Wave function ${\displaystyle f(t)=ASin\omega t}$ ${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$ ${\displaystyle T=LC}$ ${\displaystyle E(t)=ASin\omega t}$ ${\displaystyle B(t)=ASin\omega t}$ ${\displaystyle \omega ={\sqrt {\frac {1}{T}}}=C=\lambda f}$ ${\displaystyle T=\mu \epsilon }$