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Waves
Wave function
A sinusoidal wave (monochromatic) is described by
t
x
φ(x, t) = A sin 2π[ ± ]
T
λ
the sign + or − corresponds to waves moving in the direction or the opposite direction w.r.t. x, respectively.
φ(x, t) = instantaneous displacement at the time t and
in the point x (deformation of a solid, pressure in a gas,
magnetic or electric field, ...). When φ(x, t) oscillated in the
of propagation with have longitudinal waves, When φ(x, t)
oscillates in the perpendicular direction we have transversal
waves. When wave oscillates in a direction and this does
not varies we have a polarized wave.
Wave function
Waves
A sinusoidal wave (monochromatic) is described by
t
x
φ(x, t) = A sin 2π[ ± ]
T
λ
the sign + or − corresponds to waves moving in the direction or the opposite direction w.r.t. x, respectively.
A wave amplitude or maximal displacement
λ = vT = wave-length of the wave, T time periodicity
of the wave propagating at velocity v. The period T is the
time necessary to the wave to travel a length λ.
f = T1 = λc temporal frequency of the wave.
Wave intensity
I = 2π 2 f 2 vρA2
I intensity [W/m2 ] of the wave; power emitted for area unit
in the normal direction of propagation. ρ = density of the
W
medium. Decibel (dB) β = log II0 where I0 = 10−12 m
2.
Speed of sound in a�material
v=
E
ρ
v speed of the longitudinal waves in liquids or solids where
E is the bulk modulus (compressibility) of the material or
the Yung’s coefficient.
v=
�
F
µ
v speed of the transversal waves in a wire with tension F
and linear density µ [Kg/m]
Interference
Principle of superposition
If in a given point and time two or more waves coexists, the effective
displacement is the sum (resultant) of the single wave displacements.
Standing wave
For waves of equal amplitude and frequency,. propagating in opposite directions we have
φ1 = A sin 2π[
φtot
t
x
t
x
+ ] , φ2 = A sin 2π[ − ]
T
λ
T
λ
x
t
= φ1 + φ2 = 2A cos[2π ] sin[2π ]
λ
T
Harmonic systems
A harmonic system of period T (vibrating string, air
column in a tube, etc...) is characterised by a set of standing
waves with quantised spectrum of frequencies fn = Tn .
A vibrating string of length L is characterised by quantised wave-lengths λn = 2L
n , such that:
v
v
fn =
=n·
λn
2L
Harmonic systems
Fourier’s theorem
every wave (function) of fundamental period T generic
timber is uniquely characterised by the superposition of
sinusoidal (monochromatic) waves of defined ferquencies,
amplitudes and phases
φtot (t) =
�
n
t
+ ψn ]
An sin[2π
Tn
Spettro di frequenza
Ampiezza
Human voice
frequenza
Onda sonora
tempo
tempo
Doppler effect
f� = f
v ± v0
v ∓ vs
in the approximation of small velocities w.r.t. the speed
of sound v in the medium. f � = effective frequency experienced by the observer; f frequency of the sound source;
±v0 and ∓vs are the velocities of the observer and of the
sound source w.r.t. the medium, respectively (+ e − for
approaching and − e + departing).
Undulatory optics
Spectrum of electromagnetic waves
A = amplitude
I ∝ A2 = intensity
A
I
A cos θ
I cos2 θ
Interference (Young’s experiment)
The superposition of coherent electromagnetic waves
coming from two slits S1 and S2 generates on a screen an
interference patten. The directions of the maximal constructive interferences is given
S sin θ = kλ, massimi
λ
S sin θ = (2k + 1) , minimi
2
where k = 1, 2, 3, . . . , S = distance between S1 e S2 , θ angle
w.r.t. the optical axis ; λ = wave-length.
Diffraction or refraction
gratings
Huygens-Fresnel’s principle: every point on a slit behaves
as a independent source of electromagnetic waves
Thin films
In every reflection the light has a variation of phase π.
The superposition of coherent light waves coming from
the reflection on the first and second surface of a thin film
of transparent material originates on the screen interference
patten whose maximal directions are given by:
λ
d sin θ = (2k + 1) , massimi
2
d sin θ = kλ, minimi
where k = 1, 2, 3, . . . , d = is the size of the film, θ is the
angle w.r.t. the optical axis, λ is the wave length. .
OTTICA GEOMETRICA
Leggi della riflessione
θi = θr
θi e θr angolo di incidenza e di riflessione. Il raggio incidente, riflesso e la normale alla superficie giacciono sullo
stesso piano.
Leggi della rifrazione
sin θi
n2
=
= n21
�
sin θr
n1
θi e θr angolo di incidenza e di rifrazione. n1 e n2 indici di
rifrazione del mezzo 1 e 2 (proporzionali alla velocità della
luce in quel mezzo). n21 = n indice di rifrazione relativo.
Prisma di rifrazione
L’indice di rifrazione dipende dalla lunghezza d’onda in
quel materiale n21 = λλ12
Riflessione Totale
Dal passaggio ad un mezzo più rifrangente ad uno meno
rifrangente si ha la riflessione totale interana per angoli di
incidenza maggiori dell’angolo limite θL , tale che:
sin θL =
Fibra ottica
n2
= n21
n1
Specchi (o lenti) sferici
1
1
1
+
=
s1
s2
f
s1 distanza dell’oggetto del centro della lente.
s2 distanza dell’immagine dal centro della lente.
f = 1/D = distanza focale (> 0 convergenti, < 0 divergenti), D = diottrie.
m = − ss12 ingrandimento (< 0 immagine capovolta)
Immagine REALE se i raggi luminosi convergono effettivamente
nel punto immagine in modo da poter essere proiettata su di
uno schermo. Immagine VIRTUALE se è formata dai soli
prolungamenti dei raggi luminosi.
Microscopio
Cannocchiale