2 u 1-D: 3-D: x + 2 u

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1 c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience Onde 1 1 Waves 1.1 wave propagation field Field: a physical quantity (measurable, at least in principle) function of position r and time t, defined for at least one region of space and a time interval. The field exists in a whole region of space, to a difference from a point-like particle which, by its nature, is present only in one point. The behavior of the field in one point also depends on the behavior in the surrounding points. We will consider: scalar fields u = u( r, t): scalar quantities (temperature, pressure) or single components of vector quantities (mechanical displacement, electric field E, magnetic field B ) defined in a 1-D space: r = x i or in a 3-D space: r = x i + y j + z k ; vector fields u = u ( r, t) Wave equation Several fields obey an equation of the form 2 u 1 v 2 2 u t 2 = 0 1-D: 3-D: 2 u x 1 2 u 2 v 2 t = u x + 2 u 2 y + 2 u 2 z 1 2 u 2 v 2 t = 0 2 (1) which is the wave equation. The constant parameter 1 necessarily has dimension of an inverse square velocity v. Its v2 value is determined by the parameters which appear in the equations obeyed by the specific field (for the displacement field of a vibrating string, the string tension and its specific mass; for the pressure field in a gas, the gas properties, which are functions of temperature and pressure; for the electric or magnetic field, the dielectric constant and the magnetic permeability of the medium) Solutions of the 1-D scalar wave equation In the 1-D case, u = u(x, t), a solution of equation (1) can be built as follows: take any function f(s) of a single variable s; build the variable s as s = x ± vt; u(x, t) = f(x ± vt) satisfies the equation, for any twice differentiable function f, as is easily checked by substituting into the equation.

2 c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience Onde 2 Mathematics shows that any solution of eq. (1) can be written in the form u(x, t) = f(x + vt) + g(x vt). The meaning of u(x, t) = f(x ± vt) can be analyzed asking: when u(x 1, t 1 ) = u(x 2, t 2 )? Since s 1 = x 1 ± vt 1 s 2 = x 2 ± vt 2 = (x 1 + x) ± v (t 1 + t) = x 1 ± vt 1 + ( x ± v t) = s 1 + ( x ± v t) the condition u(x 1, t 1 ) = u(x 2, t 2 ) is always satisfied by s 1 = s 2 x ± v t = 0, although this solution is often not unique. x ± v t = 0 means: a) u(x, t 1 ) = u(x + x, t 1 + t) for x = v t: given u(x, t 1 ), at time (t 1 + t) the function u reproduces itself identically, but translated by x = v t (for f(x + vt)), or by x = +v t (for f(x vt)): the u(x, t) translates with velocity v towards decreasing x (for f(x + vt)) or increasing x (for f(x vt)); b) u(x 1, t) = u(x 1 + x, t + t) for t = x/v: given u(x 1, t), in the position (x 1 + x) the function u reproduces itself identically, but translated by t = v t (anticipated, for f(x + vt)), or x = +v t (delayed, for f(x vt)): again, this shows that u(x, t) translates with velocity v towards decreasing x (regressive wave, for f(x + vt)) or increasing x (progressive wave, for f(x vt)); NOTE: s = (x±vt) has dimension of length, therefore the form f(x±vt) is not fully satisfactory because a mathematical function which gives a physical quantity must have a dimensionless argument (e.g. sin ωt instead of sin t) in order to be independent from the adopted units; a form of the type f( x ± vt ), l being a fixed length, is preferable (see below). l Periodic waves Among the infinite functions f(s) which can be exploited to build a solution to the wave equation, there are the periodic functions, for which p is the period and s f(s + p) = f(s). Remaining for the moment in the notation s = x ± vt, p has dimension of length. For a periodic function the above question (when u(x 1, t 1 ) = u(x 2, t 2 )?) has, beside the answer: for s 2 = s 1, analyzed above, also the answer: for s 2 = s 1 + np, where n is any integer, thus for x ± v t = np. In particular: a) at a given time t 1 ( t = 0) u identically reproduces itself for any x = np. The period p is called wavelength λ: u(x, t) = u(x + nλ, t), n, integer. b) at a given position x 1 ( x = 0) u identically reproduces itself for any t = n (p/v). The interval (p/v) = (λ/v) has dimension of time, and is called period T : u(x, t) = u(x, t + nt ), n, integer.

3 c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience Onde 3 Therefore T = λ/v or v = λ/t and, introducing frequency ν = 1/T, we have v = λν. The wavelength λ, which characterizes the periodic wave, is usefully exploited to write the argument of the function f(x ± vt) in dimensionless form (see note above): u(x, t) = f( x ± vt λ ) = f(x λ ± v λ t) = f(x λ ± t T ) (2) Among periodic waves sinusoidal or harmonic waves have great importance: mathematics shoes that: any periodic function of period p can be written as sum of sinusoids whose periods are p itself and all its integer sub-multiples p/n (Fourier series) any function, also non periodic, can be written as sum of infinite sinusoids of infinitesimal amplitude whose periods are all the real values (Fourier integral). Therefore the knowledge of the behavior of sinusoidal waves allows us to deduce, by Fourier series or integral, the behavior of any wave. We therefore focus on sinusoidal waves, whose numerical period is 2π (the numerical period of form (2) is 1) ( u(x, t) = u M cos 2π x ± vt ) ( 2π = u M cos λ λ x ± 2π v ) λ t = u M cos (kx ± ωt) (3) or, with non null initial phase ϕ u(x, t) = u M cos (kx ± ωt + ϕ) = u M R { e i(kx±ωt+ϕ)} = R { u M e iϕ e i(kx±ωt)} = R { ũ M e i(kx±ωt)} with: k = 2π λ ω = 2πν = 2π v λ ϕ u M ũ M = u M e iϕ wavevector angular frequency initiale phase real amplitude complex amplitude A wave of the form (3), which has precisely defined values of ω and k, is called monochromatic wave. The argument (kx ± ωt + ϕ) is called the phase, ϕ being the initial phase, or phase in the origin. Remember that velocity v = λν = ω is fixed by the physics of the relevant field, therefore the k value of ω uniquely defines the value of k, and viceversa Scalar 3-D waves For scalar quantities u defined in the 3-D space u = u(x, y, z, t) the phase is no longer function of only x and t, but is function of x, y, z and t. The question therefore arises: at a given time t, which points have the same phase? The ensemble of these points (the locus of points having the same phase) is called wavefront, and the variety of solutions of the 3-D wave equation can be

4 c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience Onde 4 classified according to the shape of wavefronts. The simplest cases are the plane wave and the spherical wave. The plane wave can be given by (3) when the wavefronts are parallel to the (y, z) plane: / y = / z = 0 = u(x, y, z, t) = u(x, t) = u M cos (kx ± ωt + ϕ) ; in all the points of any plane parallel to the (y, z) plane the quantity u has the same value, and each of these x = const planes is a wavefront. For any plane wave cartesian coordinates (x, y, z ) can always be taken, in which the axis x coincides with the propagation direction. It is therefore possible to define the vector k =k i, with modulus k = 2π/λ and direction which coincides with that of x, i.e. with the propagation direction. Any plane monochromatic wave can thus be represented, in any cartesian coordinates, by its wavevector: u(x, y, z, t) = u( ( k ) { r, t) = u M cos r ωt + ϕ = R ũ M e i( k r ωt) } ; (4) ( k ) in this expression we can always indicate only the progressive wave r ωt, the regressive ( k ) wave r + ωt being represented by the same expression but with opposite k. 1.2 Standing waves Superposition of waves having opposite directions We noted above that when two or more waves are superposed the amplitudes always sum up (in general vectorially), while the intensity can be different from the sum of the intensities. We analyzed the superposition of two plane waves having same frequency and same polarization; considering the case in which their propagation directions are parallel we found and studied the interference phenomenon. Let us now consider the case in which their propagation directions are antiparallel ; the two waves are given by u 1 = u M1 cos (kx ωt) = R { u M1 e i(kx ωt)} u 2 = u M2 cos ( kx ωt + ϕ) = R { u M2 e iϕ e i( kx ωt)} and their superposition is u 1 + u 2 = R {( u M1 e ikx + u M2 e iϕ e ikx) e iωt} even though the argument of R {...} is written as the product of a function of only x (all the round parenthesis) times a function of only t (the last exponential), both functions are complex: the real part of their product is not the product of the real parts, and the extraction of the real part gives a function which depends jointly on x and t.

5 c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience Onde 5 However if the round parenthesis was a real function it could be brought out of the operator R {...}, and amplitude u would be the product of a function of only x times a function of only t (R {e iωt }). This happens when u M1 = u M2 and ϕ = 0 or ϕ = π, i.e. e iϕ = ±1; in this case u 1 + u 2 = R { ( u M1 e ikx ± e ikx) e iωt} = R {2 cos (kx) u M1 e iωt } = 2u M1 cos (kx) R {e iωt } = R {2i sin (kx) u M1 e iωt } = 2u M1 sin (kx) R { e i(ωt π/2)} and we have a wave in which the configuration (the spatial shape) of the field u remains always identical to itself, and is multiplied by a function of time which oscillates harmonically between -1 and 1. This is not a traveling wave, but in any point (any value of x) there is a fixed amplitude (between 0 and 2u M1 ) of an oscillation which is simultaneous in all the points: all the points oscillate in phase. This wave does not travel, and is called standing wave, or stationary wave. Also from an energy point of view it can be shown that intensity (in the case of the acoustic wave in the thin rod, the power exerted across a section) oscillates with a null average. The points where amplitude is null are called nodes, those where amplitude is maximum can be called antinodes. The superposition of the two waves propagating in opposite directions can be easily obtained by the reflection of a wave. The case of the vibrating string locked in one point is probably the simplest to visualize. Taking in that point the origin of x axis we have u(x = 0, t) = 0 and therefore u(x, t) = 2u M1 sin (kx) R { e i(ωt π/2)} ; a wave comes from x = +, is reflected in x = 0 and returns towards +, such that in x = 0 there is a node of the standing wave. The form proportional to 2u M1 cos (kx) is obtained if in x = 0 there is a free end of the string: the free end reflects the wave similarly to a locked end, but gives an antinode instead of a node in the reflection point. In any point of the string, of abscissa x, the incident wave passes, and the reflected wave also passes, with a phase delay determined by the path from x to 0, by a phase jump of π upon reflection, and by the path from 0 to x. The phase difference between the two waves always remains the same: where it is destructive we find a node, where it is constructive we find an antinode. In particular, the nodes are found for kx = (2π/λ) x = mπ, i.e. for x = mλ/2. If the string is locked also at its second end, at distance L from the first one, the standing wave remains the same if the second end falls on a node, i.e. if L = mλ/2. Only for wavelengths that satisfy this condition (i.e. for λ = 2L/m, corresponding to k = (2π/λ) = mπ/l) there are standing waves on the string of length L locked at its extremes. These waves have frequencies ν = ω/2π = v/λ = mv/2l, integer multiples of the fundamental frequency v/2l (which is the reciprocal of the time in which the wave performs a round trip) and, redefining the origin of times, have the form u(x, t) = 2u M1 sin (kx) R {e iωt }. 1.3 Electromagnetic waves We only discuss the kinematics of wave propagation; only in the example of the longitudinal acoustic wave in a thin rod we have analyzed a specific propagation law (the dynamics for that type of wave).

6 c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience Onde 6 In the case of the electric and magnetic fields E and B, the laws obeyed by these fields when they are time dependent (Maxwell s equations), imply that: both E and B separately obey a wave equation, with the same value of the constant 1/v 2 indicated below; the fields E and B of an e.m. wave are both perpendicular to k : the electromagnetic waves are transversal, and can thus be polarized in the ways described above; the presence of a field E of the form (??) necessarily implies the presence of a field B of the same form, and viceversa. The ensemble of the fields E and B, necessarily linked to each other, of the form (??) is called electromagnetic wave (e.m. wave); the fields E and B are also perpendicular to each other, in such a way that E B has the direction of k ; to describe the state of polarization it is sufficient to specify one of the two vectors (usually E ), the direction of the other one being automatically fixed; the moduli of the fields E and E B are in the ratio = v B In particular, in the wave equations obeyed by the fields E and B the parameter 1/v 2 is given by (1/v 2 ) = εµ. The value of v in vacuum (velocity of the electromagnetic waves in vacuum 1 or velocity of light in vacuum) is indicated by c =. In any other medium v = 1 = ε0 µ 0 εµ 1 ε0 ε r µ 0 µ r = c εr µ r = c n ; n = ε r µ r > 1 is called refractive index. Also for waves of other nature (e.g. acoustic waves) the velocity depends on the medium in which the wave propagates, and it is always possible (although seldom done) to define a refractive index as the ratio of the wave velocity in a reference medium to the velocity in a specific medium. In several media the wave velocity, and therefore the refractive index, depends on frequency; such media are called dispersive. The relation ω = ω(k) is called dispersion relation. In a non dispersive medium, such as those considered up to now, the velocity v is independent from frequency, and the dispersion relation is the simple proportionality ω = vk. For electromagnetic waves in vacuum ω = ck Energy carried by Electromagnetic waves Let us consider an electromagnetic wave: the electric and magnetic energy density is given by: ρ E = 1/2ε 0 E 2 ρ B = 1/2µ 0 B 2

7 c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience Onde 7 The total energy density (energy per unit volume)is: The energy du of a small volume dv = Acdt is: ρ EM = 1/2ε 0 E 2 + 1/2µ 0 B 2 = ε 0 E 2 (5) Hence du = ε 0 ce 2 Adt (6) 1 du A dt = P A = I = ε 0cE 2 (7) The intensity I of the wave is defined as I = P/A, power transported by the wave per unit area of the crossed section The average intensity is In the case of the harmonic wave : < I >=< ε 0 ce 2 >= ε 0 c < E 2 >= 1/2ε 0 c < E 2 0 >= ε 0 ce 2 0 (8) The most important result is the quadratic dependence of intensity on amplitude; in particular, for a harmonic wave I u 2 M = ũ M. 2. This quadratic dependence is not specific to this example, but occurs for all the phenomena which obey the wave equation (for the electromagnetic wave it can be derived from Maxwell s equations), and is one of the most important features of wave phenomena. 1.4 Superposition of waves. Interference Superposition of two or more waves The wave equation is linear: the sum of two (or more) solutions is a solution. But the intensity, being a quadratic function, is a non linear function of field. Therefore when two or more waves are superposed: the resulting field is the sum (in general, vectorial) of the fields at any time: the amplitudes are summed; the resulting intensity is NOT necessarily the sum of intensities. We analyze now the superposition of two plane waves u 1 e u 2 which have: same frequency: if the frequencies are different it can be shown that the intensity of the superposition is simply the sum of the intensities; same direction: if the directions are different the same phenomena described below occur, but with a position dependence (at least 2-D) which is more elaborated; we note that same direction and frequency imply that the wavevectors are identical, except at most for sign; let us consider first wavevectors of same sign;

8 c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience Onde 8 same polarization: two waves of identical wavevector and different polarizations form a single polarized wave (in general of elliptical polarization); it can be shown that if the two different polarizations are perpendicular the intensity of the resulting wave is independent from the phase difference among the two components, and coincides with the sum of the two intensities. As already noted (see polarization) one phase can be arbitrarily redefined by a redefinition of the origin of time, therefore only the phase differences are relevant, and we can always write u 1 = u M1 cos (kx ωt) = R { ũ M1 e i(kx ωt)} u 2 = u M2 cos (kx ωt + ϕ) = R { ũ M2 e i(kx ωt)} with ũ M1 = u M1 = I 1 = α ũ M1 2 = α u 2 M1 = u M1 = I 1 /α ũ M2 = u M2 e iϕ = u M2 (cos ϕ + i sin ϕ) = I 2 = α ũ M2 2 = α u 2 M2 = u M2 = I 2 /α The superposition has thus amplitude u 1 + u 2 = R { ũ M1 e i(kx ωt)} + R { ũ M2 e i(kx ωt)} = R { (ũ M1 + ũ M2 ) e i(kx ωt)} = R { ũ M e i(kx ωt)} and intensity I = α ũ M 2 = α ũ M1 + ũ M2 2 = α u M1 + u M2 (cos ϕ + i sin ϕ) 2 = = α (u M1 + u M2 cos ϕ) + iu M2 sin ϕ 2 = = α ( u 2 M1 + 2u M1u M1 cos ϕ + u 2 M2 cos2 ϕ + u 2 M2 sin2 ϕ ) = α (u 2 M1 + u2 M2 + 2u M1u M2 cos ϕ) = = I 1 + I I 1 I 2 cos ϕ For the phase difference ϕ among the two superposed waves there are 2 possibilities: if it is a well defined quantity, which does not vary in time, the two waves are called coherent; if instead, as it usually happens, it varies in time, either continuously or by jumps, the two waves are called non coherent. if the two waves are non coherent, cos ϕ and thus I continuously vary, and (if the variation of ϕ is not very slow) we observe only a I, further averaged over a time interval in which ϕ takes many different value. I 1 e I 2 are constants, and coincide therefore with their averages, while cos ϕ = 0; thus I = I 1 + I 2

9 c 2013 C.S. Casari - Politecnico di Milano - Introduction to Nanoscience Onde 9 if the two waves are coherent (cos ϕ, constant, coincides with cos ϕ ) I = I 1 + I I 1 I 2 cos ϕ the amplitudes are summed but the intensities are not: the intensity of the superposition is in general different from the sum of the intensities, can be larger or smaller than the sum of the intensities, and depends on the phase difference ϕ: this fact is called INTERFERENCE. In particular the intensity I can have any value between I 1 + I 2 2 I 1 I 2 = ( I1 I 2 ) 2 (for ϕ = π: waves in counter-phase, destructive interference) and I 1 + I I 1 I 2 = ( I1 + ) 2 I 2 (for ϕ = 0: waves in phase, constructive interference). In the peculiar case in which I 2 = I 1 we have I = 0 for destructive interference and I = 4 I 1 = 2 ( I 1 + I 2 ) for constructive interference. Coherence having been defined, we can note that a wave is polarized or not if its two independent transversal components are coherent or not.

Chapter 16 Waves. Types of waves Mechanical waves. Electromagnetic waves. Matter waves

Chapter 16 Waves. Types of waves Mechanical waves. Electromagnetic waves. Matter waves Chapter 16 Waves Types of waves Mechanical waves exist only within a material medium. e.g. water waves, sound waves, etc. Electromagnetic waves require no material medium to exist. e.g. light, radio, microwaves,