Oscillations, Vibrations and Waves

Simple Harmonic Motion

Summary of Time relationships in SHM

Displacement is x = rsin(wt)
Velocity is: v = wrcos(wt)
Acceleration is: a = -w²rsin(wt) =-w²x

Velocity is: v = +/- w√(r²- x²)

Acceleration is: a = -w²x

Free osciallation describes a system in which the total energy of the system does not decrease with time

A system which returns to its equilibrium position with no overshoot in the shortest possible time is said to critically damped

Wave Descriptions

Transverse waves are defined as waves in which the displacements of the particles of the medium are perpendicular to the direction of wave travel. Can be polarised
Longitudinal waves are defined as waves in which the oscillation of the particles of the medium are parallel to the direction of wave travel. Cannot be polarised.

Summary of terms associated with waves

The displacement of a particles is its instantaneous distance from the equilibrium.(s,x)
The amplitude of a wave is defined as the maximum displacement of a particle from its equilibrium.(a,r)
The wavelength of a wave is defined as the distance between any two points on adjacent cycles which are vibrating in phase.(λ)
The frequency of a wave is defined as the number of complete waves passing a given point per second.(f)

The velocity of the wave is given by: v = f λ

Interference and Diffraction

The principle of superposition states: At a point where two or more waves meet, the instantaneous displacement is the vector sum of the individual displacements due to each wave at that point. This basically tells us that at the point where the waves meet, the resulting wave will depend not only on the amplitudes of each wave but also on their phase relationship.

Constructive intereference: for two sources in phase the path difference must be equal to an even number of half-wavelengths.

Destructive intereference: for two sources in phase the path difference must be equal to an odd number of half-wavelengths.

Young's double-slit formula is: λ = ws / D

where w = distance between adjacent fringes (fringe width).
s = distance between slits (slit width).
D = distance from slits to screen.

Diffration grating formula is dsinθ = nλ

where d = grating spacing
n = the spectrum order(i.e n = 1 for the first order diffraction image).

Stationary Waves

Stationary waves are produced when two progressive waves of equal frequency and speed and nearly equal amplitude travelling in opposite direction are superimposed.

The velocity v of a transverse wave along a strecthed string is: v = √ T / μ

where T = tension in the string
μ = mass per unit length of string

Doppler Effect

The Doppler effect for moving source(s) and moving listener(L) is: f_L = (v + v_L/ v + v_s)*f_s

where v = velocity in medium -usually 340 m/s for sound

EM Spectrum

Wave Type	Wavelength Range(m)	Sources	Detectors
Radio	>0.001	Oscillating charges	Antennae and tuned circuits
Microwave	0.1-10^-5	Magnetrons and Klystrons	Antennae and tuned circuits
Infra-red	10^-4-7*10^-7	Thermal vibrations of atoms in hot bodies	Semiconductors, Photographic film
Visible	710^-7-410^-7	Energy level exchanges of electrons in atoms	Eye, Photographic film, Semidconductors
Ultra Violet	4*10^-7-10^-9	Energy level exchanges of electrons in atoms	Fluorescent chemicals, Photoelectric effect
X-rays	<5*10^-8	X ray tubes	Photographic film, Ionisation detectors
Gamma	<10^-10	Radioactive nuclei	Photographic film, Ionisation detectors

Reflection and Refraction

The law of refraction or sometimes known as Snell`s law is: n₁sinθ₁ = n₂sinθ₂

Examples of Refractive index values n

Medium Symbol Absolute Refractive Index

Air n_a 1.003
Water n_w 1.33
Ice n_i 1.30
Glass n_g 1.50
Diamond n_d 2.42

Medium	Symbol	Absolute Refractive Index
Air	n_a	1.003
Water	n_w	1.33
Ice	n_i	1.30
Glass	n_g	1.50
Diamond	n_d	2.42

Total Internal Reflection occurs when light passing from one medium to another of lower refractive index is incident on the boundary at an angle greater than the critical angle for the two media. All of the light is reflected from the boundary, and there is no transmission.

The critical angle for Total Internal Reflection is: sinθ_crit = n₁/n₂

Lenses and Telescopes

The Power of a lens is defined as P = 1 / focal length (f)

A Real image is one through which real rays of light actually pass and can therfore be formed on a screen. A Virtual image is one from which rays of light appear to have come and cannot be formed on a screen.

The lens formula for a thin lens is: 1/u + 1/v = 1/f

where u = object distance
v = image distance
f = focal length of lens

The linear magnification m is defined as: m = v / u

Resolving power-the angle of the first dark ring produced by a hole of diameter(w) with light of wavelength(λ) is: sinθ = 1.22 λ / w

Oscillations, Vibrations and Waves