Wave Attenuation Due to Losses and Transmission Across Junctions

Abstract

In large building constructions like ships, cars, houses, etc., the energy flows from a mechanical source.

Problems

5.1

Two semi-infinite beams are connected at right angels. The junction between the beams is hinged, i.e., no bending moment can be transferred from one beam to the other. A longitudinal wave is incident on the junction in beam 1. Determine the transmitted and reflected energy flows as function of the incident energy flow. The two beams are identical, width b, height h,  Young’s modulus E,  Poisson’s ratio $\nu$, and density $\rho$.

5.2

The incident wave in Problem 5.1 is a flexural wave. Determine the transmitted and reflected energy flows.

5.3

At a junction n, identical semi-infinite plates are connected along a straight line. In one of the plates, a plane flexural wave is incident on the junction (normal incidence). Determine the attenuation of the energy flow to any of the other plates. Neglect the translatory motion of the junction.

5.4

A longitudinal wave is incident on the discontinuity shown in the figure. Determine the ratio between the incident and transmitted energy flows.

5.5

Two semi-infinite beams oriented along the same axis are connected by means of an elastic interlayer as shown in the figure. A longitudinal wave is incident on the interlayer. Determine the attenuation across the junction. Consider only longitudinal waves.

5.6

A flexural wave is propagating in a beam toward a blocking mass as shown in the figure. A flexural wave is transmitted across the blocking mass. Determine the ratio between incident and transmitted energy flows. It is sufficient to define incident and transmitted waves and the boundary conditions necessary for solving the problem. Assume the blocking mass to be rigid. Its mass is M and its rotational mass moment of inertia J. The width of the beam is b and its height h.

5.7

An evanescent flexural wave on a beam is described by

$$ w(x,t) = A \cdot \exp [i(\omega t + \kappa _0 \eta x / 4) - \kappa _0 x] $$

where $\kappa _0$ is the real part of the wave number and $\eta$ the loss factor. Determine the energy flow in the beam due to this wave.

5.8

A thin infinite plate is excited at the origin by a point force $F = F_0 \cdot \exp ({i}\omega t)$. Determine the far-field displacement of the resulting flexural wave.

5.9

An infinite plate is excited by a point force. The displacement in the far-field is given by the result of Example 5.8. Neglecting the losses in the plate show that the power transmitted to the far-field is equal to power input at the excitation point.

5.10

Two semi-infinite plates are joined together along a straight line. The joint is allowed to rotate only. A flexural plane wave, unit amplitude, is incident on the junction. Determine the amplitude R of the reflected wave and show that $\left| R \right| = 1$ when no propagating wave is transmitted across the junction.

5.11

Two semi-infinite and equal plates are joined together along a straight line. The joint is allowed to rotate only. A flexural plane wave is incident on the junction. Show that the transmission coefficient for transmission across the junction is given by Eq. (5.129).

5.12

Two semi-infinite and equal plates are joined together along a straight line. The joint is allowed to rotate only. A flexural plane wave is incident on the junction. Show that the transmission coefficient for diffuse incidence is two-third of the transmission coefficient for normal incidence.