Mathematical Physics


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1 Mathematical Physics MP205 Vibrations and Waves Lecturer: Office: Lecture 19 Dr. Jiří Vala Room 1.9, Mathema<cal Physics, Science Building, North Campus Phone: (1) E Mail:
2 VIII. BOUNDARY EFFECTS AND INTERFERENCE We are now about to study effects that take place when a traveling wave encounters a barrier, or a different medium, or small obstacles. Reflection of wave pulses Recall that a given normal mode on a string with fixed ends can be regarded as the superposition of two sine waves of equal amplitude, wavelength, and frequency traveling in opposite directions. Mathematically, the following two statements are equivalent: (i) Normal mode: nπx y(x, t) = A sin cos ωt L
3 (ii) Two traveling waves: y(x, t) = A 2 sin nπx L ωt + A 2 sin nπx L + ωt At the fixed ends x = 0 and x = L we have y(0, t) = y(l, t) = A 2 sin ( ωt) + A 2 = A 2 sin (ωt) + A 2 sin (ωt) = sin (ωt) so the oppositely traveling waves must at all times produce equal and opposite displacements at the fixed ends. This condition must define the reflection process for any traveling wave when it encounters a rigid boundary.
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6 What happens if a string with a certain tension and linear density µ 1 is attached to a string with µ 2 µ 1? We will observe at the discontinuity (x = 0). reflection, and transmission Quantitatively: assuming a partial reflection and transmission y 1 (x, t) = f 1 t x + g v 1 t + x 1 v 1 y 2 (x, t) = f 2 t x v 2
7 The transverse displacement y at x = 0 must be the same for both strings, and also at each instant the strings must join with equal slopes and have equal tensions (discontinuity would lead to a large acceleration). These conditions imply y 1 (0, t) = y 2 (0, t) y 1 x (0, t) = y 2 (0, t) x f 1 (t) + g 1 (t) = f 2 (t) 1 f v 1 (t) 1 g 1 v 1 (t) 1 = 1 f v 2 (t) 2 Integrating the last equation gives v 2 f 1 (t) v 2 g 1 (t) = v 1 f 2 (t)
8 and together with f 1 (t) + g 1 (t) = f 2 (t) we can find the solutions for g 1 (t) and f 2 (t): g 1 (t) = v 2 v 1 f v 2 + v 1 (t) 1 2v f 2 (t) = 2 f v 2 + v 1 (t) 1 which describes the situation at x = 0 for all t. Important piece of reasoning: The equations above relate g 1 and f 1 and f 2 at the same value of the argument: g 1 (τ) = const. f 1 (τ) and f 2 (τ) = const. f 1 (τ) where f 1 is defined to be a function of t x/v 1, and g 1 is defined to be a function of t + x/v 1, so τ = t f x f /v 1 = t g + x g /v 1
9 For t f = t g = t we have x g = x f that is, the displacement associated with pulse g 1 at any given instant, at any value of x is directly related to the value of f 1 at the same time at the position x. Specifically g 1 t + x v 1 = v 2 v 1 f v 2 + v 1 t x 1 v 1 where the l.h.s. is the real displacement in the reflected pulse at x < 0 and r.h.s. is the displacement in the incident pulse if it had continued to x > 0. The equality above shows the reflected pulse (scaled down by v 2 v 1 v 2 +v ) is reversed right to left w.r.t. 1 the incident pulse. If v 2 < v 1, it is also turned upside down.
10 Similarly we can relate the transmitted waveform f 2 to the incident one f 1 : from τ = t x 1 v = t x 2 1 v we get x 2 2 = v 2 v x 1 1, and thus f 2 t v 2x/v 1 = 2v 2 f v 2 v 2 + v 1 t x 1 v 1 The transmitted pulse suffers not only a change in height but also a scale change along x. It should be noted that f 1 and g 1 represent physical displacements only if x 0, whereas f 2 only if x 0. Example: The development of the reflected and transmitted pulses from a given incident pulse for the case v 2 = v 1 /2
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12 Extreme cases of Eq. 1 are (a) String 2 is infinitely massive, i.e. v 2 = 0: g 1 t + x = f v 1 t x 1 v 1 (b) String 2 is massless (or absent), i.e. v 2 = : g 1 t + x = f v 1 t x 1 v 1 These cases represent reflection from a fixed end and from a loose end of the string.
13 Actual examples of the reflection and transmission of pulses traveling along stretched strings lighttoheavy spring heavytolight spring reflections from a very light thread
14 Longitudinal vs. transverse waves: polarization Several examples: a stretched string exhibits transverse vibrations, a long spring shows both transverse (shear) and longitudinal waves, providing a good analog of a real solid, and a column of liquid or gas displays longitudinal waves only if there is no resistance to a change of shape. Transverse waves allow existence of two different directions of polarization for the vibrations which are perpendicular to one another and to the direction of propagation. The different polarization states may have different speeds associated with them. For example, a crystalline medium with different interatomic distance in one direction than another. In anisotropic crystals there may be three different wave speeds  one for longitudinal and two for transverse vibrations  along one direction.
15 Also a given interface may behave differently w.r.t. longitudinal and transverse waves. For example, the interface between water and a wall acts as almost completely rigid w.r.t. longitudinal waves but as a completely free end w.r.t. transverse waves; considering a standing wave of water, the wall would represent a node for longitudinal vibrations of water and an antinode for transverse ones.
16 Waves in two dimensions For example waves on a surface. Phenomena of interest: (i) a change in direction of traveling waves, (ii) those involving the superposition of disturbances arriving at a given point from different directions. The same phenomena occur in three dimensions also but are easier to handle in 2D. We shall be dealing with various solutions to 2D wave equation: or 2 z x z y 2 = 1 2 z v 2 x 2 2 z r z r r = 1 2 z v 2 x 2
17 We will focus on two special forms of wave: (1) Plane waves or straight waves, generated by oscillations of a straight or flat object of linear dimensions which are large compared to λ; (2) Circular waves, generated by an object whose linear dimensions are small compared to λ; for large r these become straight waves. Instead of solving the equations above rigorously, we shall start with the assumption that we have straight or circular waves and will consider their behavior in various physical situations.
18 The HuygensFresnel principle If waves traveling outward from a source encounter a barrier with a tiny aperture then this aperture appears to act just like a new point source from which circular waves spread out. Since the effect of the barrier is to suppress all propagation of the original disturbance except through the aperture, it does not matter whether the original waves are straight or circular. Consider a circular wave pulse, expanding at a speed v. If the pulse is created at the origin at t = 0, then the effect occurring at r + r at time t + r/v could be ascribed to the agitation of the medium at r at time t, thus treating the disturbance as something handed on from point to adjacent point through the medium (Huygens & Fresnel).
19 It is a consequence of the HuygensFresnel principle that an unimpeded circular wave pulse gives rise to a subsequent circular wavefront and a straight pulse gives rise to a straight wavefront. However, given a circular wavefront HBGI, there will be developed from it at a later time a circular wavefront DCEF: Each point, e.g. B, gives rise to a circular wavelet KCL, and the totality of these wavelets generates a reinforcement along the line DCEF that is tangent to them all at a given instant. The shortest distance between this locus and the original wavefront is v t.
20 However, this construction would define two subsequent wavefronts: (i) a new wavefront just described above, and (ii) another wavefront moving back towards the source. Consequently, we need to incorporate a unidirectional property of a traveling wave. To do this, we require that the disturbance starting out from a given point in the medium at a given instant is not equally strong in all directions: If O is the true original source; S is the origin of a Huygens wavelet, and P is the point at which the disturbance is recorded, then the effect at P due to the region near S is a function f (θ) of the angle between OS and SP, such that f (θ) = 0 for θ = π.
21 The Huygens construction is useful but essentially qualitative, and to do more is quite difficult. Specific mathematical formulation of the Huygens principle was published by Helmholtz (1859) and was further developed by Kirchhoff (1882). This method is useful though in the analysis of the optical interference phenomena for example.
22 Reflection and refraction of plane waves As in 1D, we can expect a partial reflection and a partial transmission of waves at a boundary of two different media. But in addition we must in 2D and 3D consider also changes of direction. The simplest case: A straight wave striking a straight boundary. We can expect Snell s law as the outcome. This can be obtained by the Huygens construction. Let AA be a straight wavefront at the instant t when the point A encounters the boundary, and let BB be the position where the wavefront has advanced in the original medium at a later time.
23 Each successive point along the boundary between A and B, as it is reached by the wavefront, becomes the center of new Huygens wavelets, advancing into the second medium and traveling back into the original medium. Tangents to these wavelets will be the new wavefronts. Later, the original wavefront touches the boundary at point C : the wavelet that started from A spreading back into the original medium, will have radius AC. the line C C, tangent to this later wavelet, will also be tangent to all the wavelets arising from the point along the boundary between A and A : the line C C is a new wavefront.
24 From the geometry, the angles i and i made with the boundary by the incident and reflected wavefronts satisfy sin i = A C AC = AC AC = sin i The angle between the boundary and the wavefront equals the angle between the normal to the boundary and the normal of the wavefront (i.e. the ray direction). Thus i and i are the angles of incidence and reflection for rays encountering a straight boundary. The process of refraction is analyzed similarly: the wavelet that has advanced from A to the second medium from the time the wavefront was at AA to the time when it touches the boundary at C has the radius AC ;
25 C C is tangent to this wavelet (and to all the other at this instant) and is the wavefront in medium 2: Let v 1 and v 2 be velocities in media 1 and 2; then if the time evolved is t A C = v 1 t and AC = v 2 t. The angle of refraction r is then given as and therefore sin i = v 1 t AC and sin r = v 2 t AC sin i sin r = v 1 v 2 (1)
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27 Remarks: calculating actual amplitudes of the reflected/transmitted waves is highly nontrivial in general; longitudinal and transverse waves behave differently; with transverse waves the behavior depends on polarization; at normal incidence, i = 0, the problem is essentially onedimensional though differences between longitudinal and transverse waves behavior may remain (e.g. waterwall example above).
28 Examples of the reflection and refraction of water waves in a ripple tank: (a) Rigid boundary (b) Boundary of a region with different velocities (e.g. water depth)
29 Remarks: A change of direction in the wavefront occurs whenever the wave velocity varies with position. For example, the speed of compressional waves in gases depends on temperature (recall v T). Thus if there is a temperature gradient in a gas, the waves traveling through will be progressively bent. Also if the medium itself is in motion with different parts having different velocities refraction will occur.
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