List of Physics Oscillations Formulas, Equations Latex Code

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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
• Mechanic oscillations
• Electric oscillations
• Waves in long conductors
• Amplitude of a driven oscillation
• Coupled conductors and transformers
• Pendulums
• Angular frequency for a damped oscillation
• Displacement of a driven oscillator
• Displacement of a slightly damped oscillator
• Energy change in a damped oscillation
• Energy transmitted by a harmonic wave
• Harmonic wave function
• Kinetic energy of simple harmonic motion
• Phase constant of a driven oscillation
• Potential energy of simple harmonic motion
• Power transmitted by a harmonic wave
• Standing-wave function
• Superposition of standing waves on a string with both ends fixed
• Total energy of simple harmonic motion
• Velocity at resonance frequency of a driven oscillator

1. Oscillations and Waves

• Harmonic oscillations

  Equation

  
  
  

  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
  • 
  
  

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• Mechanic oscillations

  Equation

  
  
  

  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
  
  

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  • Oscillations of Mechanical Systems

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• Electric oscillations

  Equation

  
  
  

  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
  
  

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• Waves in long conductors

  Equation

  
  
  

  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
  
  

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• Amplitude of a driven oscillation

  Equation

  
  
  

  Latex Code

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

  Explanation

  
  

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• Coupled conductors and transformers

  Equation

  
  
  

  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 \\
              \frac{V_1}{V_2}=\frac{I_2}{I_1}=-\frac{i\omega M}{i\omega L_2+R_{\rm load}}\approx-\sqrt{\frac{L_1}{L_2}}=-\frac{N_1}{N_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
  
  

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• Pendulums

  Equation

  
  
  

  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
  
  

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• Angular frequency for a damped oscillation

  Equation

  
  
  

  Latex Code

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

  Explanation

  
  

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• Displacement of a driven oscillator

  Equation

  
  
  

  Latex Code

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

  Explanation

  
  

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• Displacement of a slightly damped oscillator

  Equation

  
  
  

  Latex Code

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

  Explanation

  
  

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• Energy change in a damped oscillation

  Equation

  
  
  

  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)
          

  Explanation

  
  

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• Energy transmitted by a harmonic wave

  Equation

  
  
  

  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
          

  Explanation

  
  

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• Harmonic wave function

  Equation

  
  
  

  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]
          

  Explanation

  
  

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• Kinetic energy of simple harmonic motion

  Equation

  
  
  

  Latex Code

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

  Explanation

  
  

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• Phase constant of a driven oscillation

  Equation

  
  
  

  Latex Code

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

  Explanation

  
  

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• Potential energy of simple harmonic motion

  Equation

  
  
  

  Latex Code

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

  Explanation

  
  

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• Power transmitted by a harmonic wave

  Equation

  
  
  

  Latex Code

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

  Explanation

  
  

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• Standing-wave function

  Equation

  
  
  

  Latex Code

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

  Explanation

  
  

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

  Equation

  
  
  

  Latex Code

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

  Explanation

  
  

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• Total energy of simple harmonic motion

  Equation

  
  
  

  Latex Code

              E_{Total} = \frac{1}{2}kA^2
          

  Explanation

  
  

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• Velocity at resonance frequency of a driven oscillator

  Equation

  
  
  

  Latex Code

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

  Explanation

  
  

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