Differential equation of simple harmonic motion or SHM,

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

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Equation of motion of SHM

$$\boxed{{\displaystyle x(t)=Asin(\omega t+\varphi )}}$$

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Differential equation of Damped harmonic oscillator

$$\boxed{{\displaystyle \frac{d^{2}x}{d{t}^2}+2\gamma \frac{dx}{dt}+{\omega }_o^{2}x=0}}$$

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Damping coefficient

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

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Angular frequency of damped oscillator

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

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Quality factor

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

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Relaxation time

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

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Energy dissipation in damped oscillator

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

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Amplitude variation in damped oscillator

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

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Differential equation of forced harmonic oscillator

$$\boxed{{\displaystyle \frac{d^{2}x}{d{t}^2}+2\gamma \frac{dx}{dt}+{\omega }_o^{2}x=f_{o}sin({\omega }_{f}t)}}$$

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Amplitude of forced harmonic oscillator

$$\boxed{{\displaystyle A=\frac{f_o}{\sqrt{({\omega }_o^2-{\omega }_f^2{)}^2+4{\gamma }^2{\omega }_f^2}}}}$$

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Differential equation of LCR circuit or electrical oscillator

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

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1-D Wave equation

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

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3-D Wave equation

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

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Solution of 1-D Wave equation

$$\boxed{{\displaystyle u(x,t)=Asin(kx\pm \omega t+\varphi )}}$$

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Frequency & Angular frequency

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

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Wavelength and Wavenumber

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

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Velocity of Wave

$$\boxed{{\displaystyle v=f\lambda orv=\frac{\omega }{k}}}$$

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Velocity of waves in stretched string

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

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Frequency of waves in stretched string

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

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

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Condition for maximum ( Constructive interference )

$$\boxed{{\displaystyle pathdifference=n\lambda }}$$

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Condition for minimum ( Destructive interference )

$$\boxed{{\displaystyle pathdifference=(2n-1)\frac{\lambda }{2}}}$$

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Cosine law ( Reflection on thin film )

$$\boxed{{\displaystyle 2\mu tcosr=n\lambda }}$$

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Bandwidth ( of Air wedge interference pattern )

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

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Diameter of thin wire ( Air wedge )

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

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Radius of nth dark ring ( Newton's Rings )

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

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Wavelength of light ( Newton's Rings )

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

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

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

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Minimum thickness ( Anti-Reflection coatings )

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

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Grating equation

$$\boxed{{\displaystyle sin\theta =nN\lambda }}$$

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Resolving power of grating

$$\boxed{{\displaystyle \frac{\lambda }{d\lambda }=n{N}^{\prime }}}$$

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Dispersive power of grating

$$\boxed{{\displaystyle \frac{d\theta }{d\lambda }=\frac{nN}{cos\theta }}}$$

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De-broglie wavelength

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

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Position - Momentum Uncertainty principle.

$$\boxed{{\displaystyle \mathrm{\Delta }x\mathrm{\Delta }P=\hslash }}$$ In terms of velocity v of a particle, $$\boxed{{\displaystyle \mathrm{\Delta }xm\mathrm{\Delta }v=\hslash }}$$

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Position - Momentum Uncertainty principle.

$$\boxed{{\displaystyle \mathrm{\Delta }E\mathrm{\Delta }t=\hslash }}$$

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1D Schrodinger's Equation ( time dependent )

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

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3D Schrodinger's Equation ( time dependent )

$$\boxed{{\displaystyle (-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V)\mathrm{\Psi }=i\hslash \frac{\partial \mathrm{\Psi }}{\partial t}}}$$

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Time independent schrodinger's equation

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

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Energy of particle in 1 dimensional square well potential or box

$$\boxed{{\displaystyle E_n=\frac{n^2{h}^2}{8m{L}^2}}}$$

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Loudness

$$\boxed{{\displaystyle L=lo{g}_{10}(I)}}$$

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Sound intensity level (SIL)

$$\boxed{{\displaystyle SIL=lo{g}_{10}(\frac{I}{I_o})}}$$

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Absorption

$$A={\alpha }_{floor}{S}_{floor}+{\alpha }_{ceiling}{S}_{ceiling}$$ $$+{\alpha }_{walls}{S}_{walls}+...$$

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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 $${\alpha }_{av}=\frac{A}{S}$$

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Sabine's formula

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

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Frequency of ferromagnetic rod

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

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Frequency of Piezoelectric crystal

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

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Numerical Aperture ( NA )

$$\boxed{{\displaystyle NA=sin{\theta }_a}}$$

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Numerical Aperture ( NA )

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

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Fractional/Relative refractive index change $\mathrm{\Delta }$

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

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