VARIATIONAL PRINCIPLE APPLIED TO THE PROPAGATION OF ACOUSTIC WAVES. LATERAL WAVES: Conical acoustic waves at an interface

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1 A. La Rosa Lecture Notes ALIED OTIC VARIATIONAL RINCILE ALIED TO THE ROAGATION OF ACOTIC WAVE LATERAL WAVE: Conical acoustic s at an interface The figure shows an XY plane interface. A source at is located along the z axis in the, for example, air medium. At t=0 the source emits spherical s advancing at speed (while travelling in the medium 1), and reaching a denser medium 2 (water or stainless steel) where the sound propagates faster at speed. [For example, in air = 340 m/s; in water = 1443 m/s; in stainless steel =5980 m/s]. 1 is located at an aritrary distance from the interface ( need not e large compared with the length ). Air t olid or liquid Fig.1 ource, located at a distance from the interface, emits spherical fronts (starting at t=0). The reflected spherical front can e otained y assuming a fictitious source located at (the latter is a mirror image of.) The points (on the reflected front) and (on the refracted front) are reached simultaneously at the time t. The rays, and satisfy the nell s law.

2 Drawing the direct fronts For a given time t, we draw a spherical surface with radius t with center at, as shown in Fig.1. This constitutes the front of the direct at time t. Drawing the ordinary reflected fronts Let s consider an aritrary point reached y the ray in a time t. ( and satisfy the nell s law). The path is travelled in a time t with speed. A collection of points reached y reflected rays at the time t constitutes a front of the ordinary reflected front. Below we descrie a simple geometrical procedure to draw such a reflected front. Let s recall that when we applied the principle of least time to rays reflected from the interface (ection 11.2), we concluded that the trace of the actual reflected ray could e otained as if it were originated from a fictitious source located at (where was located at the mirror image of ). That is, in Fig.1 aove, the ray can e considered eing an image of the ray (i.e. oth travel at the same speed ). Therefore, the distance is equal to t. Accordingly, the ordinary front can e otained y drawing a sphere of radius t with center at, as shown in Fig.2 elow. As an aid to the eye (and facilitate the drawings of the direct and reflected fronts when starting from scratch), a point L has een drawn at the interface (see Fig.2 elow). That point is reached simultaneously at the time t y a ray emitted from and y a ray emitted from the fictitious source. The direct front can then e drawn as a sphere with radius L with center at ; the ordinary reflected front can e drawn as a sphere with radius L with center at. (at time t ) t L Fig.2 front drawn as a sphere with radius L with center at. front drawn as a sphere with radius L with center at.

3 Drawing the refracted fronts Aritrarily we chose a value of numerically equal to. First, consider first a ray that is incident normal to the interface, and choose a total travel time is equal to, for example, 2 seconds (1 s travelling in the water with speed, 1 sec travelling in the solid with speed.) This determines one point of the refracted front. Then, for any other incident ray like M (hitting the surface at a given distance x M from the origin,) determine the time of travel t M. ince the total travel time is 2 seconds, the length of the ray M is (2 t M ). The orientation of M (which depends on the orientation of M, is determined y the nell s law. Repeating this process for different points M, the refracted front can e otained. ee schematics in Fig.3 elow. Detailed on scale construction of the refracted front is left as an assignment. -front Water -front latimun M (at time t) Fig. 3 uggested diagram for the construction of the refracted front. Extraordinary reflected fronts In contrast to the one point reflection rays addressed in the sections aove (that allowed constructing the ordinary reflected front), this time consider alternative reflection paths, like those paths that partially propagate in medium 1 and partially propagate along the interface. More precisely, consider an aritrary path that leaving, reaches the interface at X, propagates along the interface with speed until it reaches an (aritrary) fixed point W, and then goes to. This new alternative path is shown in Fig.4

4 Water olid X front W Wave Fig. 4 Alternative to the ordinary reflect ray, the extraordinary ray XW are also considered as potential path for reflection. To simplify the prolem, let s reak it down into two partial analysis. For example, let s consider first the following variational prolem (see Fig.5). For aritrary fixed points and, what is the location of the point X that makes the time of travel an extreme? Water latinum x X L Fig. 5 etting for finding the particular path that makes the OL(X) an extreme. Formally, what we have to do is to find the stationary values of the function, X X t(x) = = t(x) c1 c2 which leads to the following result: the path X is an extreme when the angle of incidence satisfies sin ( extreme c ) c 1 2

5 That is, X is an extreme when its angle of incidence is equal to the total internal reflection angle a o, which is given y s in ( ) c / c o 1 2. One can argument then that, for a given fixed point, the path that makes the optical path length stationary is M 1 M 2 indicated in Fig.6. This ray reaches the point at a time t that is lower than t. Water front (at a time t) t t olid M 1 M 2 Fig. 6 Illustration of the path M 1 M 2 that makes stationary the travelling time from and. Conical fronts If point is reached y the ray M 1 M 2 in a given time t, let s find out other points in the medium-1 that are reached y a ray M 1 M 2 at the same time t. It turns out, the locus of such points lie along a line that a) passes through the given point, and ) makes an angle 0 with the horizontal. (This can e demonstrated using the similarity properties of triangles, and it is left as an assignment.) Hint: Notice, the difference etween t(m 1 M 2 ) and t(m 1 M 2 ) is given y, t = (M 2 )/ (M 2 M 2 )/ (M 2 )/. sing the similarity properties of triangles demonstrate that this difference is identically equal to zero). Notice, these locus of point,,, constitute a front of different shape than the ordinary spherical reflected front (also drawn in Fig. 7 for comparison). Thes fronts are known a conical s.

6 Water olid M 1 front t t' M 2 Fig. 7 Illustration of the generation of conical s. t' M 2 (at the time t) Conical front M 1