## List of Physics Oscillations Formulas, Equations Latex Code

rockingdingo 2023-03-26 #physics #oscillations #wave 1 0

List of Physics Oscillations Formulas, Equations Latex Code

In this blog, we will introduce most popuplar formulas in Oscillations, Physics. We will also provide latex code of the equations. Topics include harmonic oscillations, mechanic oscillations, electric oscillations, waves in long conductors, coupled conductors and transformers, pendulums, harmonic wave, etc.

### 1. Oscillations and Waves

• #### Harmonic oscillations

##### Latex Code
            \Psi(t)=\hat{\Psi}(t)e^{i(\omega t \pm \phi)}=\hat{\Psi}(t)\cos (\omega t \pm \phi) \\
\sum_{i} \hat{\Psi_{i}}\cos(\alpha_{i} \pm \omega t) =\hat{\Phi}\cos (\beta \pm \omega t) \\
\tan (\beta)=\frac{\sum_{i} \hat{\Psi_{i}} \sin (\alpha_{i})}{\sum_{i} \hat{\Psi_{i}} \cos (\alpha_{i})} \\
\hat{\Phi}^{2} = \sum_{i}  \hat{\Psi_{i}^{2}} + 2 \sum_{j > i} \sum_{i} \hat{\Psi_{i}} \hat{\Psi_{j}} \cos (\alpha_{i} - \alpha_{j}) \\
\int x(t) dt=\frac{x(t)}{i \omega} \\
\frac{d^{n}(x(t))}{d t^{n}}=(i \omega)^{n} x(t)

##### Explanation

Latex code for the harmonic oscillations. I will briefly introduce the notations in this formulation.

• : Amplitude
• Superposition of more harmonic oscillations with the same frequency

• #### Mechanic oscillations

##### Latex Code
            m\ddot{x}=F(t)-k\dot{x}-Cx \\
F(t)=\hat{F}\cos(\omega t) \\
-m\omega^2 x=F-Cx-ik\omega x \\
\omega_0^2=C/m \\
x=\frac{F}{m(\omega_0^2-\omega^2)+ik\omega} \\
\dot{x}=\frac{F}{i\sqrt{Cm}\delta+k} \\
\delta=\frac{\omega}{\omega_0}-\frac{\omega_0}{\omega} \\
Z=F/\dot{x} \\
Q=\frac{\sqrt{Cm}}{k}

##### Explanation

Latex code for the Mechanic Oscillations. I will briefly introduce the notations in this formulation.

• : Construction of spring with constant
• : Damping constant
• : Periodic force
• : Velocity
• : Impedance of the system
• : The quality of the system
• Velocity resonance frequency: The frequency with minimal

• #### Electric oscillations

##### Latex Code
            \text{Impedance} \\
Z=R+ix \\
\text{Series connection} \\
V=IZ, Z_{\rm tot}=\sum_i Z_i~,~~L_{\rm tot}=\sum_i L_i~,~~ \frac{1}{C_{\rm tot}}=\sum_i\frac{1}{C_i}~,~~Q=\frac{Z_0}{R}~,~~ Z=R(1+iQ\delta) \\
\text{Parallel connection} \\
\frac{1}{Z_{\rm tot}}=\sum_i\frac{1}{Z_i}~,~~ \frac{1}{L_{\rm tot}}=\sum_i\frac{1}{L_i}~,~~ C_{\rm tot}=\sum_i C_i~,~~Q=\frac{R}{Z_0}~,~~ Z=\frac{R}{1+iQ\delta}

##### Explanation

Latex code for the Electric oscillations. I will briefly introduce the notations in this formulation.

• : Phase Angle
• : Impedance of a Resistor
• : Capacitor
• : Self inductor
• : Quality of a coil

• #### Waves in long conductors

##### Latex Code
             Z_0=\sqrt{\frac{dL}{dx}\frac{dx}{dC}} \\
v=\sqrt{\frac{dx}{dL}\frac{dx}{dC}}

##### Explanation

Latex code for the Waves in Long conductors. I will briefly introduce the notations in this formulation.

• : is transmission velocity

• #### Amplitude of a driven oscillation

##### Latex Code
            A = \frac{{F_0 }}{{\sqrt {m^2 \left( {\omega _0^2 - \omega ^2 } \right)^2 + b^2 \omega ^2 } }}


• #### Coupled conductors and transformers

##### Latex Code
            M_{12}=M_{21}:=M=k\sqrt{L_1L_2}=\frac{N_1\Phi_1}{I_2}=\frac{N_2\Phi_2}{I_1}\sim N_1N_2 \\
\Phi_{12}=M_{12}I_2 \\
\Phi_{21}=M_{21}I_1

##### Explanation

Latex code for Coupled conductors and transformers. I will briefly introduce the notations in this formulation.

• : part of the flux originating from I_{2{} through coil 2, which is enclosed by coil 1
• : coefficients of mutual induction
• : Coupling factor

• #### Pendulums

##### Latex Code
            T=1/f \\
T=2\pi\sqrt{m/C} \\
T=2\pi\sqrt{I/\tau} \\
T=2\pi\sqrt{I/\kappa} \\
T=2\pi\sqrt{l/g}

##### Explanation

Latex code for Coupled conductors and transformers. I will briefly introduce the notations in this formulation.

• : Oscillating spring
• : Physical pendulum
• : Torsion pendulum
• : Mathematical pendulum

• #### Angular frequency for a damped oscillation

##### Latex Code
            \omega ' = \omega _0 \sqrt {1 - \left( {\frac{b}{{2m\omega _0 }}} \right)^2 } = \omega _0 \sqrt {1 - \frac{1}{{4Q^2 }}}


• #### Displacement of a driven oscillator

##### Latex Code
            x = A\cos \left( {\omega t + \delta } \right)


• #### Displacement of a slightly damped oscillator

##### Latex Code
            x = A_0 \exp \left( { - \frac{b}{{2m}}t} \right)\cos \left( {\omega 't + \delta } \right)


• #### Energy change in a damped oscillation

##### Latex Code
            \frac{{\Delta E}}{E} = - \frac{b}{m}T \\
E = E_0 \exp \left( { - \frac{b}{m}t} \right) = E_0 \exp \left( { - \frac{t}{\tau }} \right)


• #### Energy transmitted by a harmonic wave

##### Latex Code
            \Delta E = \frac{1}{2}\mu \omega ^2 A^2 \Delta x = \frac{1}{2}\mu \omega ^2 A^2 \upsilon \Delta t


• #### Harmonic wave function

##### Latex Code
            y(x,t) = A\sin \left[ {2\pi \left( {\frac{x}{\lambda } - \frac{t}{T}} \right)} \right] = A\sin \left[ {k(x - \upsilon t)} \right]


• #### Kinetic energy of simple harmonic motion

##### Latex Code
            K = \frac{1}{2}kA^2 \sin ^2 \left( {\omega t + \delta } \right)


• #### Phase constant of a driven oscillation

##### Latex Code
            \tan \delta = \frac{{b\omega }}{{m\left( {\omega _0^2 - \omega ^2 } \right)}}


• #### Potential energy of simple harmonic motion

##### Latex Code
            U = \frac{1}{2}kA^2 \cos ^2 \left( {\omega t + \delta } \right)


• #### Power transmitted by a harmonic wave

##### Latex Code
            P = \frac{{dE}}{{dt}} = \frac{1}{2}\mu \omega ^2 A^2 \upsilon


• #### Standing-wave function

##### Latex Code
            y(x,t) = A_n \cos (\omega _n t + \delta _n )\sin (k_n x)


• #### Superposition of standing waves on a string with both ends fixed

##### Latex Code
            y(x,t) = \sum\limits_n {A_n \cos (\omega _n t + \delta _n )\sin (k_n x)}


• #### Total energy of simple harmonic motion

##### Latex Code
            E_{Total} = \frac{1}{2}kA^2


• #### Velocity at resonance frequency of a driven oscillator

##### Latex Code
            \upsilon = + A\omega \cos \left( {\omega t} \right) = - A\omega \sin \left( {\omega t - \frac{\pi }{2}} \right)