Waves in Solids

Abstract

The energy flow in structures is caused by various types of waves. The description and understanding of the character of these waves are therefore essential. In this chapter, the basic differential equations governing the one-dimensional propagation of longitudinal, transverse, torsional, and bending waves are discussed.

Problems

3.1

An infinite beam is oriented along the x-axis in a coordinate system. The displacement along the x-axis is $\xi = A \cdot \sin (\omega t - k_l x)$, where $k_l$ is the wave number for quasi-longitudinal waves. The width of the beam is b and its height h. Determine the displacement perpendicular to the x-axis of the beam. Assume that $\sigma _y$ and $\sigma _z$ are equal to zero in the beam.

3.2

Determine the resulting kinetic energy in the beam of Problem 3.1. Consider only the effects due to quasi-L-waves.

3.3

An L-wave is propagating in an infinite and homogeneous beam oriented along the x-axis of a coordinate system. The resulting displacement is defined by $f(x - c_l t)$. Determine the kinetic and potential energies plus the energy flow due to this wave.

3.4

A semi-infinite and homogeneous beam with constant cross section area S is oriented along the x-axis of a coordinate system. At $x = 0$ the beam is excited by a force F(t) in the direction of the positive x-axis. Determine the displacement in the beam. Consider only L-waves. As an example let the force be given by $F(t) = F_0 \sin \omega t$.

3.5

Torsional waves are propagating in an infinite cylindrical and homogeneous shaft with radius R. Due to the wave motion the torsional angle ${\varTheta }$ varies as ${\varTheta } = {\varTheta } _0 \sin (k_t x - \omega t)$. Determine the potential and kinetic energies per unit length of the shaft as well as the energy flow in the shaft which is oriented along the x-axis of a coordinate system.

3.6

Flexural waves are propagating in an infinite and homogeneous beam oriented along the x-axis of a coordinate system. The displacement of the beam is given by w(x, t). Determine the potential energy per unit length of the beam based on the general expression Eq. (3.17) and the definition of the strain in Eq. (3.72). Neglect shear effects.

3.7

The deflection $\eta$ of an infinite and homogeneous string oriented along the x-axis is at $t = 0$ equal to $\eta (x,0) = \cos (\pi x / L)$ for $- L / 2 < x < L / 2$ otherwise zero.

The string is at rest at $t = 0$. Determine the displacement of the string when it is released at $t = 0$. Neglect the losses.

3.8

A thin, infinite, and homogeneous beam is oriented along the x-axis in a coordinate system. The mass per unit length is ${m}^{\prime }$ and its bending stiffness ${D}^{\prime }$. For $t < 0$ the beam is at rest having the lateral displacement $\exp [ - (x / 2a)^2]$. The beam is released at $t = 0$. Determine the displacement of the beam for $t > 0$. Compare the discussion in Sect. 3.8.

3.9

An attempt is made to measure the energy flow in a thin homogeneous beam by means of just one accelerometer. The material and geometrical parameters of the beam are known.

The bending stiffness and wavenumber are denoted ${D}^{\prime }$ and $\kappa$. Losses are neglected. In the first case the lateral displacement of the beam, which is oriented along the x-axis of a coordinate system, is equal to $w(x,t) = A \cdot \exp [i(\omega t - \kappa x)]$. Determine the energy flow in the beam as function of the time average of the velocity squared measured at the point $x = x_0$.

In the second case the near field can not be neglected. The displacement is $w(x,t) = A \,\cdot \, \exp (i\omega t) \,\cdot \, [\exp ( - i\kappa x) - i \,\cdot \, \exp ( - \kappa x)]$. Determine the ratio between the actual energy flow and the energy flow estimated by means of the velocity squared measured by means of the accelerometer at the point $x = x_0$.

In the third case the near field but not a reflected field can be neglected. The displacement is given by $w(x,t) = A \cdot \exp (i\omega t) \cdot [\exp ( - i\kappa x) + X \cdot \exp (i\kappa x)]$. Again calculate the ratio between the actual and measured energy flows at the point $x = x_0$.

3.10

Show that the bending moment per unit length induced by shear in an orthotropic plate is given by Eq. (3.132) as

$$ {M}^{\prime }_{xy} = - \sqrt{D_x D_y } \cdot (1 - \sqrt{\nu _x \nu _y }) \cdot \dfrac{\partial ^2w}{\partial x\partial y}. $$

The plate is oriented in the x-y-plane of a coordinate system.

3.11

An L-wave is propagating in an infinite beam oriented along the x-axis of a coordinate system. The displacement is $\xi (x,t) = A \cdot \exp [i(\omega t - k_l x)]$. Show that the time average of the energy flow ${\Pi }$ is , where is the time average of the total energy per unit length of the beam and $c_l$ the phase velocity of the wave.

3.12

Show that the intensity of L-waves propagating in the beam of Problem 3.11 is given by $I_x = - \sigma _x \cdot \partial \xi / \partial t$ where $\xi$ is the displacement in the beam. Start by considering the total energy per unit volume of the beam.