Capítulo 4. Trastornos de la conducta alimentaria y trastornos de la personalidad
4.4. Comorbilidad entre los trastornos de la conducta alimentaria y los trastornos la
Very often, the refractive index will depend on the frequency of the oscil-lation. This means that, in turn, the phase velocity will also depend on frequency. Hence, the components of the general solution to the wave equation, Eq. (3.48), will tend to drift apart - a phenomenon known as dis-persion.
With no unique phase velocity, the question of the how fast a wave signal actually propagates naturally arises. Consider the general solution wave expression given by Eq. (3.48). We may think of this as describing a
3.7. GROUP VELOCITY 61 wave packet, which must travel with some average velocity. We call this the group velocity vg(k0)associated with some average wavevector k0. 3.7.1 Derivation of the group velocity
If the domain of k does not exceed by too much some average wavevec-tor k0, then to a good approximation we can write ωk as a Taylor series expanded to first order
ωk≈ ωk0+ ∇kωk0· (k − k0) , (3.49) where ∇kis the gradient operator with respect to wavevector
∇k = ˆx ∂
∂kx + ˆy ∂
∂ky + ˆz ∂
∂kz (3.50)
and ∇kωk0 is evaluated at k = k0. Substituting Eq. (3.49) into Eq. (3.48) we obtain
E (r, t) = exp [i (∇kωk0 · k0− ωk0) t]
Z
Ekexp [ik · (r − ∇kωk0t)] d3k
= eiθ(t)Ψ(z(r, t), 0), (3.51)
where
θ(t) = (∇kωk0· k0− ωk0) t (3.52) and
z(r, t) = r − ∇kωk0t. (3.53) Hence E (r, t) is modulated by a time-dependent phase eiθ(t) whilst be-ing translated with a velocity ∇kωk0. This is the group velocity . Dropping the ‘0’ subscript on k, this is
vg(k) = ∇kωk. (3.54)
3.7.2 Significance of the group velocity
What, then, is the physical significance of the group velocity? The question is of particular importance in optics when, under certain conditions, the phase velocity may exceed the speed of light c. However, Einstein’s Special Theory of Relativity tells us that nothing can travel faster than the speed of light. Does this not imply a contradiction?
In fact, the answer is no. To be precise, the Special Relativity tells us that no physical signal may travel faster than c. This means that any-thing ‘traveling’ faster than c cannot be carrying any meaningful information, which might be re-interpreted as saying that nothing is really traveling at this speed. Only physically observable quantities such as energy, momentum or charge can transmit information. The group velocity of a wave-packet is then the speed at which such observables may be transmitted. The phase velocities of the component parts of the wave-packet should then be viewed more in terms of a mathematical framework describing the physical phe-nomenon, rather than an observable part of the physics.
3.8 Summary
• The simple harmonic oscillator – The ideal spring
The equation of motion for an ideal spring of stiffness constant Ksuspending a mass m is
md2u
dt2 = −Ku. (3.55)
This has the general solution
u (t) = A sin (ωt + φ) , (3.56) where
ω = K m
1/2
(3.57) is the angular frequency and φ is a phase factor determining the position at time t = 0.
– Energy in a simple harmonic oscillator
The total energy of an simple harmonic oscillation with amplitude Ais
= 12KA2. (3.58)
• The wave equation
The general form of the wave equation is
3.8. SUMMARY 63
∇2u = n2 c2
∂2u
∂t2, (3.59)
where n is the refractive index and c is the phase velocity of a simple harmonic wave.
• Polarisation – Unpolarised
which means that the displacement is randomly orientated in space.
– Longitudinal polarisation
In this case, the displacement is in the propagation direction. As an example, sound waves in fluids are longitudinal. However, electromagnetic waves are generally not.
– Transverse polarisation
In this case, the displacement is perpendicular to the propa-gation direction. This is generally the case for electromagnetic waves. Moreover, the transverse polarisation may be
∗ linearly polarised
in which the displacement always lies in the same plane, or
∗ elliptically polarised
in which the polarisation rotates around the direction of prop-agation.
• Properties of a medium – Linear medium
In a linear medium, wave propagation in the medium may be accurately modeled by a linear differential equation.
– Isotropic medium
In an isotropic medium, the material properties are the same in every direction. This means that the refractive index will not depend on direction of propagation or the orientation of the po-larisation. Conversely, materials that do have a directional de-pendence are known as anisotropic.
– Homogeneous medium
In a homogeneous medium, the material properties the same ev-erywhere. That is, they do not depend on spatial position. Con-versely, a medium in which the properties do depend on spatial position is called inhomogeneous.
• The wavefront
Points in space having the same phase within their cycle of oscillation constitute a wavefront.
– The speed of the wave front is then the phase velocity v
– The distance between successive wavefronts in the direction of propagation is the wavelength λ
– The magnitude of the wavevector |k| is related to the wavelength via
|k| = k = 2π
λ. (3.60)
In an isotropic medium k is perpendicular to the wavefront.
– The phase velocity is related to the wavevector via
v = ω
k. (3.61)
• Plane waves
A plane wave solution of the wave equation may be written as
E (r, t) = E0ei(k·r−ωt), (3.62) allowing us to write a general solution in the form
E (r, t) = Z
Ekei(k·r−ωkt)d3k. (3.63)
• Group velocity
The group velocity is given by
vg(k) = ∇kωk. (3.64)
This is the speed at which a physical signal may be transmitted.
Part II
Wave Optics
65
4. The Huygens-Fresnel Principle
4.1 General remarks
Following the philosophy of Chapter 3, we continue with our exploration of general wave phenomena without necessitating any recourse to the phys-ical specifics of the waves. In this pursuit, the general features of wave propagation may be derived from a handful of simple concepts.
In the interests of simplicity, we confine ourselves to media that are lin-ear, isotropic and (locally) homogeneous. Here, we briefly remind ourselves of the meanings of these terms as introduced in the Chapter 3:
• Linearity
In a linear medium, wave propagation in the medium may be ac-curately modeled by a linear differential equation. In particular, the propagation of the disturbance depends only on the amplitude of the oscillations. Higher order terms may be considered negligible.
• Isotropy
In an isotropic medium, the material properties are the same in every direction. This means that the refractive index will not depend on direction of propagation or the orientation of the polarisation.
• Homogeneity
In a homogeneous medium, the material properties, and hence the refractive index, are the same everywhere. That is, they do not de-pend on spatial position.
The last point concerning homogeneity will need a little modification, since we shall be concerned with propagation between media of different refrac-tive index, which could be interpreted as an inhomogeneous system. How-ever, such cases will be restricted to a sharp discontinuity in the refractive indices. Within specific regions, the refractive index (for a given frequency) stays constant. Hence, we shall describe this situation as locally homoge-neous.
The first principle we shall apply is due to the Dutch physicist Christiaan Huygens (1629 - 1695). Huygens’ Principle [1] explains wave propagation in terms of spherical wavelets radiating from all points on a wavefront. At some later time, a new wavefront may be constructed from the sum of these
67
wavelets. We therefore begin this chapter with a consideration of spherical waves, so that we may better understand the concepts to follow.
Huygens’ Principle may be thought of as more of a rule of thumb than a law of physics. Its principal merit is that it generally works, although there are a few situations that it fails to describe adequately, such as the backwards propagation of the spherical wavelets. The reason for this lack in explanatory power comes from the neglect of interference effects, which truly characterise wave phenomena (although, in some texts, interference effects are included in the definition of Huygens’ Principle.
Our understanding of interference effects comes largely from the work of the French physicist Augustin-Jean Fresnel (1788 - 14 July). When this theory is combined with Huygens’ Principle, we have a truly powerful ex-planatory tool to hand in the form of the Huygens-Fresnel Principle [2].
From this principle, the general principles of wave propagation may be de-rived. As in Chapter 3, we again emphasise that these principles are not restricted to optical propagation but have universal application throughout physics where-ever wave phenomena is encountered.
4.2 Learning objectives
The aims of this section are to gain understanding of
• The formulation of spherical waves
• Huygens’ Principle
• Interference and coherence
• The Huygens-Fresnel Principle
• Application of these principles to find:
– The Law of Rectilinear Propagation – The Law of Reflection
– The Law of Refraction (Snell’s Law)