Differential equation of simple harmonic motion or SHM,

$$\boxed{\frac{d^2 x}{dt^2}+\omega^2 x = 0}$$

---

Equation of motion of SHM

$$ \boxed{x(t)=A\;sin\;(\omega t + \phi) }$$

---

Differential equation of Damped harmonic oscillator

$$\boxed{\frac{d^2 x}{dt^2}+2 \gamma \frac{dx}{dt}+\omega_o^2 x = 0}$$

---

Damping coefficient

$$\boxed{\gamma = \frac{b}{2m}}$$

---

Angular frequency of damped oscillator

$$\boxed{\omega=\sqrt{\omega_o^2 - \gamma^2}}$$

---

Quality factor

$$\boxed{Q=\frac{\omega}{2 \gamma }}$$

---

Relaxation time

$$\boxed{\tau=\frac{1}{2 \gamma }}$$

---

Energy dissipation in damped oscillator

$$\boxed{E(t)=E_o\;e^{-2 \gamma t}}$$

---

Amplitude variation in damped oscillator

$$\boxed{A(t)=A_o\;e^{- \gamma t}}$$

---

Differential equation of forced harmonic oscillator

$$\boxed{\frac{d^2 x}{dt^2}+2 \gamma \frac{dx}{dt}+\omega_o^2 x = f_o\;sin\;(\omega_f \;t)}$$

---

Amplitude of forced harmonic oscillator

$$\boxed{A= \frac{f_o}{\sqrt{(\omega_o^2 - \omega_f^2)^2 + 4 \gamma^2 \omega_f^2}}}$$

---

Differential equation of LCR circuit or electrical oscillator

$$\boxed{\frac{d^2 q}{dt^2}+\frac{R}{L}\frac{dq}{dt}+\frac{q}{LC}=\frac{V_o}{L}\;sin\;(\omega t)\;}$$

---

1-D Wave equation

$$\boxed{\frac{\partial^2 u}{\partial x^2} =\frac{1}{v^2} \; \frac{\partial^u}{\partial t^2}}$$

---

3-D Wave equation

$$\boxed{\nabla^2 u =\frac{1}{v^2} \; \frac{\partial^u}{\partial t^2}}$$

---

Solution of 1-D Wave equation

$$\boxed{u(x,t)=A\;sin\;(kx \pm \omega t + \phi)}$$

---

Frequency & Angular frequency

$$\boxed{f=\frac{\omega}{2 \pi}}$$

---

Wavelength and Wavenumber

$$\boxed{\lambda=\frac{2 \pi}{k}}$$

---

Velocity of Wave

$$\boxed{v=f\;\lambda\;\;or\;\;v=\frac{\omega}{k}}$$

---

Velocity of waves in stretched string

$$\boxed{v =\sqrt{ \frac{T}{\mu}}}$$

---

Frequency of waves in stretched string

$$\boxed{f =\frac{n}{2l}\sqrt{ \frac{T}{\mu}}}$$ when n = 1 , fundamental frequency:

$$\boxed{f_{fund} =\frac{1}{2l}\sqrt{ \frac{T}{\mu}}}$$

---

Condition for maximum ( Constructive interference )

$$\boxed{path\;difference =n\;\lambda}$$

---

Condition for minimum ( Destructive interference )

$$\boxed{path\;difference =(2n-1)\frac{\lambda}{2}}$$

---

Cosine law ( Reflection on thin film )

$$\boxed{2 \mu t \;cos\;r =n\;\lambda}$$

---

Bandwidth ( of Air wedge interference pattern )

$$\boxed{\beta =\frac{\lambda}{2 \theta }}$$ if a medium of refractive index \( \mu \) is filled between glass plates, $$\boxed{\beta =\frac{\lambda}{2 \mu \theta}} $$

---

Diameter of thin wire ( Air wedge )

$$\boxed{d =\frac{l\;\lambda}{2 \beta }}$$

---

Radius of nth dark ring ( Newton's Rings )

$$\boxed{r_n =\sqrt{Rn\lambda}}$$

---

Wavelength of light ( Newton's Rings )

$$\boxed{\lambda =\frac{D_{n+k}^2 - D_n^2}{4kR}}$$ if a liquid of refractive index \( \mu \) is inserted between lens and glass plate, $$\boxed{\lambda =\frac{D_{n+k}^2 - D_n^2}{4 \mu kR}}$$ or refractive index, $$\boxed{\mu =\frac{D_{n+k}^2 - D_n^2}{4kR \lambda}}$$

---

Refractive index of liquid inserted between lens and plate ( Newton's Rings )

$$\boxed{\mu =\frac{D_{n+k}^2 - D_n^2}{d_{n+k}^2 - d_n^2}}$$

---

Minimum thickness ( Anti-Reflection coatings )

$$\boxed{t =\frac{\lambda}{4 \mu}}$$

---

Grating equation

$$\boxed{sin\;\theta = n N \lambda}$$

---

Resolving power of grating

$$\boxed{\frac{\lambda}{d \lambda}= n N'}$$

---

Dispersive power of grating

$$\boxed{\frac{d\theta}{d \lambda}= \frac{n N}{cos\;\theta}}$$

---

De-broglie wavelength

$$\boxed{\lambda = \frac{h}{p} = \frac{h}{mv}}$$ In terms of energy E of a particle, $$ \boxed{\lambda = \frac{h}{\sqrt{2mE}}} $$

---

Position - Momentum Uncertainty principle.

$$\boxed{\Delta x \;\Delta P = \hbar}$$ In terms of velocity v of a particle, $$ \boxed{\Delta x \;m\;\Delta v = \hbar} $$

---

Position - Momentum Uncertainty principle.

$$\boxed{\Delta E \;\Delta t = \hbar}$$

---

1D Schrodinger's Equation ( time dependent )

$$\boxed{-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} \Psi}{\partial x^{2}}=i \hbar \frac{\partial \Psi}{\partial t}} $$

---

3D Schrodinger's Equation ( time dependent )

$$\boxed{\Big(-\frac{\hbar^{2}}{2 m}\;\nabla^{2} + V\Big)\;\Psi=i \hbar \frac{\partial \Psi}{\partial t}} $$

---

Time independent schrodinger's equation

$$\boxed{\frac{\partial^{2} \Psi}{\partial x^{2}} + \frac{2m}{\hbar^2}\;(E-V)\Psi = 0}$$

---

Energy of particle in 1 dimensional square well potential or box

$$\boxed{\;\; E_{n}=\frac{n^{2} h^{2}}{8 m L^{2}}\;\;}$$

---

Loudness

$$\boxed{L = log_{10}\;(I)}$$

---

Sound intensity level (SIL)

$$\boxed{SIL = log_{10}\Big(\frac{I}{I_o}\Big)}$$

---

Absorption

$$A=\alpha_{floor}\;S_{floor} +\alpha_{ceiling}\;S_{ceiling}$$ $$+\alpha_{walls}\;S_{walls}+...$$

---

Average absorption coefficient

$$\alpha_{av}=\frac{\alpha_{1}\;S_{1} +\alpha_{2}\;S_{2} \alpha_{3}\;S_{3}+...}{S_1+S_2+S_3+...}$$ or $$ \alpha_{av}=\frac{A}{S}$$

---

Sabine's formula

$$\boxed{T=\frac{0.163\;V}{A}}$$

---

Frequency of ferromagnetic rod

$$\boxed{f=\frac{n}{2l}\;\sqrt{\frac{T}{\mu}}}$$ when n = 1 Fundamental frequency; $$\boxed{f=\frac{1}{2l}\;\sqrt{\frac{Y}{\rho}}}$$

---

Frequency of Piezoelectric crystal

$$\boxed{f=\frac{n}{2l}\;\sqrt{\frac{T}{\mu}}}$$ when n = 1 Fundamental frequency; $$\boxed{f=\frac{1}{2l}\;\sqrt{\frac{Y}{\rho}}}$$

---

Numerical Aperture ( NA )

$$\boxed{NA=sin\;\theta_a}$$

---

Numerical Aperture ( NA )

$$\boxed{NA=\sqrt{n_1^2-n_2^2}}$$ If fibre is inside a medium of refractive index, \( \mu\) ; $$\boxed{NA=\frac{\sqrt{n_1^2-n_2^2}}{\mu}}$$

---

Fractional/Relative refractive index change \( \Delta \)

$$\boxed{\Delta=\frac{n_1-n_2}{n_1}}$$ It can be related to numerical aperture NA as ; $$\boxed{NA=n_1\sqrt{2 \Delta}}$$

---