# MULTIDISCIPLINARY PRACTICES FOR FIRST ENGINEERING LEVELS: COLLECTION AND MATHEMATICAL TREATMENT OF TOPOGRAPHIC DATA OBTAINED WITH ULTRASONIC TECHNIQUES

The coordination between the different disciplines belonging to the same year of the degree studies is one of the main goals in the new strategies of teaching. Especially important is the case of the first year, where the students are dealing with the study of basic subjects, such as mathematics or physics. The cross-coordination between subjects is most necessary in these moments in order to establish the basic knowledge needed to deal successfully with the upper grades. This cross-coordination increases the interest of the students in other subjects useful for the Geomatic and Topography Degree.

In this paper we present a case of coordinated practices between physics and mathematics subjects, involving the obtaining of experimental data and its subsequent mathematical treatment in order to draw conclusions about the most appropriate technique decision to the problem raised. Thus, the proposed practice has two well-defined steps. The first step is related to the obtaining of experimental data following a physical technique, and the second one consists in the treatment, the proper ordination and subsequent interpretation of these data defining adequate mathematical algorithms and using a commercial mathematical program.

The experimental data obtained are related to the analysis of the phase variation of ultrasonic signals. This technique is included in a set of ultrasound techniques normally used to characterize the physical properties of materials. In short, the phase analysis gives us detailed information about the relief of surfaces (z height) with an accuracy of tenths of a millimeter, and this information can be used in several fields as in architecture, public works or even in the authentication of artworks. All these applications make the ultrasonic phase analysis an important technique to be known by the students of several engineering. Therefore, students have to learn how the experimental system works and how to obtain the necessary measures for the development of the practice. For this purpose, we propose here the obtaining of experimental data about the real relief of a piece of wall made with bricks joined with concrete. These data will be obtained using an automated position system that provides a highly accurate XYZ movement.

The second step of the practice is the data treatment. To do that, we have used Mathematica, commercial software widely used in Technical Schools. We can get a first three-dimensional representation of the object from the data matrix obtained. This representation shows us the areas which are necessary to pay attention. This suggests a detailed study into sections by vertical planes perpendicular of the OX and OY axis.
Using this software the students have to answer several questions about the shape of the piece of wall considered, as the higher difference of heights in the area analyzed, its location, if the sample has slope and its direction, etc.

Note that the study of sections of a surface is essential in first course: it helps us to represent quadric surfaces in Algebra and to understand the level curves (that are obtained as an intersection of the surface with a horizontal plane at a particular height) in Calculus, and their contour diagrams. These graphs provide one way of visualizing functions of two variables. All this has immediate applications in several matters related to the Digital Terrain Model and Topography.