In general, adaptable after-effects in solid materials1 are guided by the boundaries of the media in which they propagate. An access to guided beachcomber propagation, broadly acclimated in concrete acoustics, is to seek sinusoidal solutions to the beachcomber blueprint for beeline adaptable after-effects accountable to abuttals altitude apery the structural geometry. This is a archetypal eigenvalue problem.
Waves in plates were a part of the aboriginal guided after-effects to be analyzed in this way. The assay was developed and appear in 19172 by Horace Lamb, a baton in the algebraic physics of his day.
Lamb's equations were acquired by ambience up ceremonial for a solid bowl accepting absolute admeasurement in the x and y directions, and array d in the z direction. Sinusoidal solutions to the beachcomber blueprint were postulated, accepting x- and z-displacements of the form
\xi = A_x f_x(z) e^{i(\omega t - kx)} \quad \quad (1)
\zeta = A_z f_z(z) e^{i(\omega t - k x)} \quad \quad (2)
This anatomy represents sinusoidal after-effects breeding in the x administration with amicableness 2π/k and abundance ω/2π. Displacement is a action of x, z, t only; there is no displacement in the y administration and no aberration of any concrete quantities in the y direction.
The concrete abuttals action for the chargeless surfaces of the bowl is that the basic of accent in the z administration at z = +/- d/2 is zero. Applying these two altitude to the above-formalized solutions to the beachcomber equation, a brace of appropriate equations can be found. These are:
\frac{\tan(\beta d / 2)} {\tan(\alpha d / 2)} = - \frac {4 \alpha \beta k^2} {(k^2 - \beta^2)^2}\ \quad \quad \quad \quad (3)
and
\frac{\tan(\beta d / 2)} {\tan(\alpha d / 2)} = - \frac {(k^2 - \beta^2)^2} {4 \alpha \beta k^2}\ \quad \quad \quad \quad (4)
where
\alpha^2 = \frac{\omega^2}{c_l^2} - k^2 \quad \quad \text{and}\quad\quad \beta^2 = \frac{\omega^2}{c_t^2} - k^2.
Inherent in these equations is a accord amid the angular abundance ω and the beachcomber amount k. Numerical methods are acclimated to acquisition the appearance acceleration cp = fλ = ω/k, and the accumulation acceleration cg = dω/dk, as functions of d/λ or fd. cl and ct are the longitudinal beachcomber and microburst beachcomber velocities respectively.
The band-aid of these equations aswell reveals the absolute anatomy of the atom motion, which equations (1) and (2) represent in all-encompassing anatomy only. It is begin that blueprint (3) gives acceleration to a ancestors of after-effects whose motion is balanced about the midplane of the bowl (the even z = 0), while blueprint (4) gives acceleration to a ancestors of after-effects whose motion is antisymmetric about the midplane. Figure 1 illustrates a affiliate of anniversary family.
Lamb’s appropriate equations were accustomed for after-effects breeding in an absolute bowl - a homogeneous, isotropic solid belted by two alongside planes aloft which no beachcomber activity can propagate. In formulating his problem, Lamb bedfast the apparatus of atom motion to the administration of the bowl accustomed (z-direction) and the administration of beachcomber advancement (x-direction). By definition, Lamb after-effects accept no atom motion in the y-direction. Motion in the y-direction in plates is begin in the alleged SH or shear-horizontal beachcomber modes. These accept no motion in the x- or z-directions, and are appropriately commutual to the Lamb beachcomber modes. These two are the alone beachcomber types which can bear with straight, absolute beachcomber fronts in a bowl as authentic above.
Waves in plates were a part of the aboriginal guided after-effects to be analyzed in this way. The assay was developed and appear in 19172 by Horace Lamb, a baton in the algebraic physics of his day.
Lamb's equations were acquired by ambience up ceremonial for a solid bowl accepting absolute admeasurement in the x and y directions, and array d in the z direction. Sinusoidal solutions to the beachcomber blueprint were postulated, accepting x- and z-displacements of the form
\xi = A_x f_x(z) e^{i(\omega t - kx)} \quad \quad (1)
\zeta = A_z f_z(z) e^{i(\omega t - k x)} \quad \quad (2)
This anatomy represents sinusoidal after-effects breeding in the x administration with amicableness 2π/k and abundance ω/2π. Displacement is a action of x, z, t only; there is no displacement in the y administration and no aberration of any concrete quantities in the y direction.
The concrete abuttals action for the chargeless surfaces of the bowl is that the basic of accent in the z administration at z = +/- d/2 is zero. Applying these two altitude to the above-formalized solutions to the beachcomber equation, a brace of appropriate equations can be found. These are:
\frac{\tan(\beta d / 2)} {\tan(\alpha d / 2)} = - \frac {4 \alpha \beta k^2} {(k^2 - \beta^2)^2}\ \quad \quad \quad \quad (3)
and
\frac{\tan(\beta d / 2)} {\tan(\alpha d / 2)} = - \frac {(k^2 - \beta^2)^2} {4 \alpha \beta k^2}\ \quad \quad \quad \quad (4)
where
\alpha^2 = \frac{\omega^2}{c_l^2} - k^2 \quad \quad \text{and}\quad\quad \beta^2 = \frac{\omega^2}{c_t^2} - k^2.
Inherent in these equations is a accord amid the angular abundance ω and the beachcomber amount k. Numerical methods are acclimated to acquisition the appearance acceleration cp = fλ = ω/k, and the accumulation acceleration cg = dω/dk, as functions of d/λ or fd. cl and ct are the longitudinal beachcomber and microburst beachcomber velocities respectively.
The band-aid of these equations aswell reveals the absolute anatomy of the atom motion, which equations (1) and (2) represent in all-encompassing anatomy only. It is begin that blueprint (3) gives acceleration to a ancestors of after-effects whose motion is balanced about the midplane of the bowl (the even z = 0), while blueprint (4) gives acceleration to a ancestors of after-effects whose motion is antisymmetric about the midplane. Figure 1 illustrates a affiliate of anniversary family.
Lamb’s appropriate equations were accustomed for after-effects breeding in an absolute bowl - a homogeneous, isotropic solid belted by two alongside planes aloft which no beachcomber activity can propagate. In formulating his problem, Lamb bedfast the apparatus of atom motion to the administration of the bowl accustomed (z-direction) and the administration of beachcomber advancement (x-direction). By definition, Lamb after-effects accept no atom motion in the y-direction. Motion in the y-direction in plates is begin in the alleged SH or shear-horizontal beachcomber modes. These accept no motion in the x- or z-directions, and are appropriately commutual to the Lamb beachcomber modes. These two are the alone beachcomber types which can bear with straight, absolute beachcomber fronts in a bowl as authentic above.
No comments:
Post a Comment