Oscillations

Differential equation of simple harmonic motion or SHM,

d2xdt2+ω2x=0


x : Displacement(m)

ω : Angular frequency

Equation of motion of SHM

x(t)=Asin(ωt+ϕ)


A : Amplitude

ϕ : Initial phase

Differential equation of Damped harmonic oscillator

d2xdt2+2γdxdt+ωo2x=0


γ : Damping coefficient

ωo : Natural frequency

x : Displacement

t : time

Damping coefficient

γ=b2m


b : Damping constant

m : mass

Angular frequency of damped oscillator

ω=ωo2γ2


γ : Damping coefficient

ωo : natural angular frequency of oscillator

Quality factor

Q=ω2γ


γ : Damping coefficient

ω : Angular frequency of damped oscillator

Relaxation time

τ=12γ


γ : Damping coefficient

Energy dissipation in damped oscillator

E(t)=Eoe2γt


γ : Damping coefficient

E(t) : Energy at time t

Eo : Initial energy or undamped value

Amplitude variation in damped oscillator

A(t)=Aoeγt


γ : Damping coefficient

A(t) : Amplitude at time t

Ao : Initial Amplitude of oscillator

Differential equation of forced harmonic oscillator

d2xdt2+2γdxdt+ωo2x=fosin(ωft)


γ : Damping coefficient

fo : Amplitude of external periodic force

ωf : Angular frequency of external periodic force or external driving frequency

Amplitude of forced harmonic oscillator

A=fo(ωo2ωf2)2+4γ2ωf2


γ : Damping coefficient

fo : Amplitude of external periodic force

ωf : Angular frequency of external periodic force or external driving frequency

Differential equation of LCR circuit or electrical oscillator

d2qdt2+RLdqdt+qLC=VoLsin(ωt)


q : Charge

R : Resistance

L : Inductance

C : Capacitance

Vo : Voltage Amplitude

ω : Angular frequency of power source

Waves

1-D Wave equation

2ux2=1v2ut2


u : Displacement of particles in medium

v : velocity of wave

3-D Wave equation

2u=1v2ut2


2 : Laplacian operator

v : velocity of wave

Solution of 1-D Wave equation

u(x,t)=Asin(kx±ωt+ϕ)


u : Displacement of particles in medium

A : Amplitude of wave

ω : Angular frequency of wave

k : Wave number or wave vector

x : Displacement of wave

ϕ : Initial phase

Frequency & Angular frequency

f=ω2π


f : frequency

ω : Angular frequency

Wavelength and Wavenumber

λ=2πk


k : Wave number

λ : Wavelength

Velocity of Wave

v=fλorv=ωk


k : Wave number

λ : Wavelength

f : frequency

ω : Angular frequency

Velocity of waves in stretched string

v=Tμ


T : Tension on string

μ : Linear mass density of string

Frequency of waves in stretched string

f=n2lTμ when n = 1 , fundamental frequency:

ffund=12lTμ


T : Tension on string

μ : Linear mass density of string

l : length of string

n : number of loops formed in string

Interference

Condition for maximum ( Constructive interference )

pathdifference=nλ


n = 1,2,3....

λ : Wavelength

Condition for minimum ( Destructive interference )

pathdifference=(2n1)λ2


n = 1,2,3....

λ : Wavelength

Cosine law ( Reflection on thin film )

2μtcosr=nλ


μ = Refractive index of thin film

λ : Wavelength

t = Thickness of thin film

r : Angle of refraction

Bandwidth ( of Air wedge interference pattern )

β=λ2θ if a medium of refractive index μ is filled between glass plates, β=λ2μθ


θ = Angle of wedge

λ : Wavelength

Diameter of thin wire ( Air wedge )

d=lλ2β


β = Bandwidth of interference pattern

l : distance between point of contact and wire

λ = Wavelength of incident light

Radius of nth dark ring ( Newton's Rings )

rn=Rnλ


R = Radius of curvature of lens

n : order of the ring

λ = Wavelength of incident light

Wavelength of light ( Newton's Rings )

λ=Dn+k2Dn24kR if a liquid of refractive index μ is inserted between lens and glass plate, λ=Dn+k2Dn24μkR or refractive index, μ=Dn+k2Dn24kRλ


Dn+k = Diameter of (n+k)th dark ring

Dn = Diameter of nth dark ring

R = Radius of curvature of lens

k = Difference in order of ring

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

μ=Dn+k2Dn2dn+k2dn2


Dn+k = Diameter of (n+k)th dark ring when air is between lens and glass plate

Dn = Diameter of nth dark ring when air is between lens and glass plate

dn+k = Diameter of (n+k)th dark ring when liquid of refractive index μ is between lens and glass plate

dn = Diameter of nth dark ring when liquid of refractive index μ is between lens and glass plate

Minimum thickness ( Anti-Reflection coatings )

t=λ4μ


μ = Refractive index of thin film

λ = Wavelength of incident light

Diffraction

Grating equation

sinθ=nNλ


θ = Angle of diffraction

n = Order of spectrum

λ = Wavelength of incident light

N = Number of lines per unit length in grating

Resolving power of grating

λdλ=nN


λ = Wavelength of incident light

N = Total number of lines in grating

n = Order of spectrum

Dispersive power of grating

dθdλ=nNcosθ


λ = Wavelength of incident light

N = Number of lines per unit length of grating

n = Order of spectrum

θ = Angle of diffraction

Quantum Mechanics

De-broglie wavelength

λ=hp=hmv In terms of energy E of a particle, λ=h2mE


h = Plank's constant = 6.63×1023J.s

p = Momentum of particle

m = Mass of particle

v = Velocity of particle

Position - Momentum Uncertainty principle.

ΔxΔP= In terms of velocity v of a particle, ΔxmΔv=


Δx = Uncertainty in position

Δp = Uncertainty in momentum

= Reduced Plank's constant = h2π

Δv = Uncertainty in velocity

Position - Momentum Uncertainty principle.

ΔEΔt=


ΔE = Uncertainty in energy measurement

Δt = Life time of energy state E

= Reduced Plank's constant = h2π

1D Schrodinger's Equation ( time dependent )

22m2Ψx2=iΨt


Ψ = Wavefunction

m = mass

3D Schrodinger's Equation ( time dependent )

(22m2+V)Ψ=iΨt


Laplacian,2=2x2+2y2+2z2

Time independent schrodinger's equation

2Ψx2+2m2(EV)Ψ=0


Ψ = Wavefunction

V = Potential

Energy of particle in 1 dimensional square well potential or box

En=n2h28mL2


En = Energy of nth level

h = Plank's constant = 6.63×1023J.s

m = Mass of the particle

L = Width of the box

Acoustics

Loudness

L=log10(I)


L = Loudness of sound

I = Intensity of sound

Sound intensity level (SIL)

SIL=log10(IIo)


Io = Reference intensity or threshold of minimum ( 1012W/m2 )

I = Intensity of sound

Absorption

A=αfloorSfloor+αceilingSceiling +αwallsSwalls+...


α = Absorption coefficient

S = Surface area of each component

Average absorption coefficient

αav=α1S1+α2S2α3S3+...S1+S2+S3+... or αav=AS


A = Total bsorption

S = Total surface area

Sabine's formula

T=0.163VA


T = Reverberation time

V = Volume of hall

A = Total absorption

Ultrasonics

Frequency of ferromagnetic rod

f=n2lTμ when n = 1 Fundamental frequency; f=12lYρ


l = length of ferromagnetic rod

Y = Young's modulus

ρ = Density

Frequency of Piezoelectric crystal

f=n2lTμ when n = 1 Fundamental frequency; f=12lYρ


l = Thickness of crystal

Y = Young's modulus

ρ = Density

Fibre Optics

Numerical Aperture ( NA )

NA=sinθa


θa = Acceptance angle

Numerical Aperture ( NA )

NA=n12n22 If fibre is inside a medium of refractive index, μ ; NA=n12n22μ


n1 = Refractive index of core

n2 = Refractive index of cladding

Fractional/Relative refractive index change Δ

Δ=n1n2n1 It can be related to numerical aperture NA as ; NA=n12Δ


n1 = Refractive index of core

n2 = Refractive index of cladding