The wave equation for stiff strings and piano tuning

We study the wave equation for a string with stiffness. We solve the equation and provide a uniqueness theorem with suitable boundary conditions. For a pinned string we compute the spectrum, which is slightly inharmonic. Therefore, the widespread scale of 12 equal divisions of the just octave is not the best choice to tune instruments like the piano. Basing on the theory of dissonance, we provide a way to tune the piano in order to improve its consonance. A good solution is obtained by tuning a note and its fifth by minimizing their beats.Peer ReviewedPostprint (published version

Això sona bé: una teoria matemática de la consonància

"...Els experiments sobre percepció de la consonància de Plomp i Levelt, als anys seixanta del segle XX, juntament amb unes hipòtesis matemàtiques senzilles, permeten explicar de manera natural els intervals consonants de la musica occidental."Factoria FM

Music and mathematics. From Pythagoras to fractals

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Matemots

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Matemots

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Calibración de parámetros de infiltración y rugosidad con un modelo numérico para riego con surcos cerrados

Se presenta la calibración de los parámetros que intervienen en las ecuaciones de infiltración de Green y Ampty de rugosidad de Manning, a través del empleo de un modelo hidrodinámico completo en diferencias finitas para riego con surcos cerrados que resuelve las ecuaciones de Saint-Venant. Se muestran los resultados obtenidos en tres ensayos realizados en un surco cerrado de 50 m y, en un apéndice, el ajuste de la curva de avance en un surco cerrado de 150 m. Se concluye que el modelo numérico es aceptable para reproducir las tres fases del riego con surcos cerrados (avance, llenado y receso), por lo que puede ser empleado en el proceso de diseño para riego con surcos cerrados

Constraint algorithm for k-presymplectic Hamiltonian systems. Application to singular field theories

The k-symplectic formulation of field theories is especially simple, since only tangent and cotangent bundles are needed in its description. Its defining elements show a close relationship with those in the symplectic formulation of mechanics. It will be shown that this relationship also stands in the presymplectic case. In a natural way, one can mimick the presymplectic constraint algorithm to obtain a constraint algorithm that can be applied to k-presymplectic field theory, and more particularly to the Lagrangian and Hamiltonian formulations of field theories defined by a singular Lagrangian, as well as to the unified Lagrangian-Hamiltonian formalism (Skinner--Rusk formalism) for k-presymplectic field theory. Two examples of application of the algorithm are also analyzed

Erratum: constraint algorithm for singular field theories in the k k -cosymplectic framework

We are indebted to Prof. Dieter Van den Bleeken (Bo˘gazi¸ci University) for having drawn our attention to the error that gave rise to this note. We acknowledge the financial support from the Spanish Ministerio de Ciencia, Innovaci´on y Universidades project PGC2018-098265-B-C33, and the Secretary of University and Research of the Ministry of Business and Knowledge of the Catalan Government project 2017-SGR-932.Peer ReviewedPostprint (author's final draft

Skinner–Rusk formalism for k-contact systems

© 2022 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In previous papers, a geometric framework has been developed to describe non-conservative field theories as a kind of modified Lagrangian and Hamiltonian field theories. This approach is that of k-contact Hamiltonian systems, which is based on the k-symplectic formulation of field theories as well as on contact geometry. In this work we present the Skinner–Rusk unified setting for these kinds of theories, which encompasses both the Lagrangian and Hamiltonian formalisms into a single picture. This unified framework is specially useful when dealing with singular systems, since: (i) it incorporates in a natural way the second-order condition for the solutions of field equations, (ii) it allows to implement the Lagrangian and Hamiltonian constraint algorithms in a unique simple way, and (iii) it gives the Legendre transformation, so that the Lagrangian and the Hamiltonian formalisms are obtained straightforwardly. We apply this description to several interesting physical examples: the damped vibrating string, the telegrapher’s equations, and Maxwell’s equations with dissipation terms.We acknowledge the financial support from the Spanish Ministerio de Ciencia, Innovación y Universidades project PGC2018-098265-B-C33 and the Secretary of University and Research of the Ministry of Business and Knowledge of the Catalan Government project 2017–SGR–932Peer ReviewedPostprint (published version

Constraint algorithm for singular field theories in the k-cosymplectic framework

The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of k-cosymplectic geometry. Since these field theories are singular, we need to introduce the notion of k-precosymplectic structure, which is a generalization of the k-cosymplectic structure. Next k-precosymplectic Hamiltonian systems are introduced in or- der to describe singular field theories, both in Lagrangian and Hamiltonian formalisms. Finally, we develop a constraint algorithm in order to find a sub- manifold where the existence of solutions of the field equations is ensured. The case of affine Lagrangians is studied as a relevant example.Peer ReviewedPostprint (author's final draft