Interaction Between Longitudinal and Transverse Waves

Abstract

The simple differential equations, describing the bending of plates and beams derived in the previous chapter, are only valid as long as the structures can be considered as “thin.”

Problems

4.1

A T-wave is for $x < 0$ traveling in a thin semi-infinite plate. The plate is in the x–y-plane. The wave is traveling towards a straight edge at $x = 0$. The angle of incidence is $\beta$. The impedance of the edge is infinite. Determine the relative amplitudes of the reflected L- and T-waves at the edge.

4.2

Two semi-infinite plates are oriented in the x–y plane. The junction between the plates is defined by the line $x = 0$. Plate 1 has the thickness h and plate 2 the thickness H. An L-wave is in plate 1 traveling towards the junction. The angle of incidence is $\alpha$. Determine the ratio between the incident power and the power transmitted to plate 2.

4.3

Use Eq. (4.51) to determine the wavenumber for traveling and evanescent bending waves in a plate with thickness h. Include second-order terms. Determine also the energy flow due to a plane traveling bending wave in the plate. Include second-order terms in h.

4.4

Use Eq. (4.49) to determine the wavenumber for quasi-longitudinal waves traveling in a plate. Include only terms of the first order as the plate thickness approaches zero.

4.5

Determine the low and high frequency limits for the wave number describing flexural waves propagating in a sandwich plate. Geometrical and material parameters are given in the table of Sect. 4.10.

4.6

A bending wave, w(x, t) is propagating in a plate. Use Eq. (4.56) to show that the resulting bending moment per unit width of the plate is $- D\partial ^2w / \partial x^2$ and the corresponding shear force $- D\partial ^3w / \partial x^3$. The plate is oriented in the x–y plane.

4.7

A bending wave, $w(x,t) = \eta _0 \exp [i(\omega t - \kappa x)]$ is propagating in a plate with the thickness h. Determine the intensity in the plate. Use Eq. (4.56) in combination with the definition of the intensity. The plate is oriented in the x–y plane.

4.8

Determine the shear stress in a plate with thickness h as function of the distance y from the neutral plane of the plate. Use the result (4.44).

4.9

The wavenumber $k_x$ for a wave propagating along a so called Timoshenko beam is in Eq. (4.32) given as

$$ k_x = \pm \sqrt{\frac{1}{2}\left[ {(k_{l}^2 + k_\mathrm{t}^2 / T_\mathrm{b})\pm \sqrt{4\kappa ^4 + (k_{l}^2 - k_\mathrm{t}^2 / T_\mathrm{b})^2} } \right] }. $$

In the high frequency limit $k_x$ should approach $k_\mathrm{r}$, the wavenumber for Rayleigh waves. Determine the coefficient $T_\mathrm{b}$ for

$$ \mathop {\lim }\limits _{\omega \rightarrow \infty } k_x = k_\mathrm{r}. $$

4.10

According to Sect. 4.6 a Rayleigh wave propagating along the x-axis in a semi-infinite solid can for $y \leqslant 0$ be described by the potentials

$$ \phi = B_1 \exp [\alpha (y + h / 2)]\exp [i(\omega t - k_\mathrm{r} x)] / 2 $$

$$ {{\psi }}_z = C_2 \exp [\beta (y + h / 2)]\exp [i(\omega t - k_\mathrm{r} x)] / 2, $$

where

$$ C_2 / B_1 = i \cdot \exp [h / 2(\alpha - \beta )][k_\mathrm{r}^2 - k_0^2 (1 + \nu )] / (k_\mathrm{r} \beta ) $$

and $k_\mathrm{r}$ is the wavenumber for Rayleigh waves. The parameters $\alpha$ and $\beta$ are

$$ \beta = \sqrt{k_\mathrm{r}^2 - k_\mathrm{t}^2 },\quad \alpha = \sqrt{k_\mathrm{r}^2 - k_{l}^2 }. $$

Show that $\sigma _y = 0$ and $\tau _{xy} = 0$ for $y = 0$, i.e., on the surface of the semi-infinite solid.

4.11

Indicate a procedure to determine the intensity induced by a Rayleigh wave traveling in a semi-infinite solid. Use Eq. (4.68).