Waves. A sinusoidal wave (monochromatic) is described by. φ(x, t) =A sin 2π[ t T ± x λ ]

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1 Waves Wave function A sinusoidal wave (monochromatic) is described by φ(x, t) =A sin 2π[ t T ± x 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.

2 Wave function Waves A sinusoidal wave (monochromatic) is described by φ(x, t) =A sin 2π[ t T ± x 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. TheperiodT is the time necessary to the wave to travel a length λ. f = 1 T = c λ temporal frequency of the wave.

3 Wave intensity I =2π 2 f 2 vρa 2 I intensity [W/m 2 ] of the wave; power emitted for area unit in the normal direction of propagation. ρ = density of the medium. Decibel (db) β = log I I 0 where I 0 = W 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 = v speed of the transversal waves in a wire with tension F and linear density µ [Kg/m] F µ

4 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 φ 1 = A sin 2π[ t T + x λ ], φ 2 = A sin 2π[ t T x λ ] φ tot = φ 1 + φ 2 =2A cos[2π x λ ]sin[2π t T ]

5 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 f n = n T. A vibrating string of length L is characterised by quantised wave-lengths λ n = 2L n, such that: f n = v λ n = n v 2L

6 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 φ tot (t) = n A n sin[2π t T n + ψ n ]

7 Onda sonora Spettro di frequenza Human voice Ampiezza frequenza tempo tempo Doppler effect f = f v ± v 0 v v s 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; ±v 0 and v s are the velocities of the observer and of the sound source w.r.t. the medium, respectively (+ e for approaching and e + departing).

8 Undulatory optics Spectrum of electromagnetic waves A = amplitude I A 2 =intensity A I A cos θ I cos 2 θ

9 Interference (Young s experiment) The superposition of coherent electromagnetic waves coming from two slits S 1 and S 2 generates on a screen an interference patten. The directions of the maximal constructive interferences is given S sin θ = kλ, massimi S sin θ =(2k + 1) λ 2, minimi where k =1, 2, 3,..., S = distance between S 1 e S 2, 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

10 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: d sin θ =(2k + 1) λ 2, massimi 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..

11 OTTICA GEOMETRICA Leggi della riflessione θ i = θ r θ i e θ r angolo di incidenza e di riflessione. Il raggio incidente, riflesso e la normale alla superficie giacciono sullo stesso piano. Leggi della rifrazione sin θ i sin θ r = n 2 n 1 = n 21 θ i e θ r angolo di incidenza e di rifrazione. n 1 e n 2 indici di rifrazione del mezzo 1 e 2 (proporzionali alla velocità della luce in quel mezzo). n 21 = n indice di rifrazione relativo.

12 Prisma di rifrazione L indice di rifrazione dipende dalla lunghezza d onda in quel materiale n 21 = λ 1 λ 2 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 θ L, tale che: Fibra ottica sin θ L = n 2 n 1 = n 21

13 Specchi (o lenti) sferici 1 s s 2 = 1 f s 1 distanza dell oggetto del centro della lente. s 2 distanza dell immagine dal centro della lente. f =1/D = distanza focale (> 0 convergenti, < 0 divergenti), D = diottrie. m = s 1 s 2 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.

14 Microscopio Cannocchiale