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## Über dieses Buch

This text considers the problem of the dynamic fluid-structure interaction between a finite elastic structure and the acoustic field in an unbounded fluid-filled exterior domain. The exterior acoustic field is modelled through a boundary integral equation over the structure surface. However, the classical boundary integral equation formulations of this problem either have no solutions or do not have unique solutions at certain characteristic frequencies (which depend on the surface geometry) and it is necessary to employ modified boundary integral equation formulations which are valid for all frequencies. The particular approach adopted here involves an arbitrary coupling parameter and the effect that this parameter has on the stability and accuracy of the numerical method used to solve the integral equation is examined. The boundary integral analysis of the exterior acoustic problem is coupled with a finite element analysis of the elastic structure in order to investigate the interaction between the dynamic behaviour of the structure and the associated acoustic field. Recently there has been some controversy over whether or not the coupled problem also suffers from the non-uniqueness problems associated with the classical integral equation formulations of the exterior acoustic problem. This question is resolved by demonstrating that .the solution to the coupled problem is not unique at the characteristic frequencies and that it is necessary to employ an integral equation formulation valid for all frequencies.

## Inhaltsverzeichnis

### 1. Introduction

Abstract
The problem of the interaction between the vibration of a finite elastic structure and the associated acoustic field in an unbounded exterior domain occurs in many areas of mathematical physics. In particular, in the field of underwater acoustics it is required to determine the acoustic field either radiated by a submerged vibrating elastic structure or scattered by a submerged elastic structure. Here, the impedance mis-match between the structure and the acoustic medium is less than that between the same structure and air and hence it is often not appropriate to assume that the surface of the structure is perfectly rigid. Examples of this problem range from that of finding the radiated or scattered sound field around entire ships and submarines to that of determining the behaviour of individual sonar transducers. A practical problem considered here is to determine the acoustic field radiated by a piezoelectric ring sonar transducer and the frequency for which the maximum response is obtained.
Siamak Amini, Paul John Harris, David T. Wilton

### 2. Integral Equation Formulations of the Exterior Helmholtz Problem

Abstract
In this chapter various integral equation formulations for the Helmholtz equation in the infinite region exterior to a bounded three-dimensional structure are considered. In later chapters the most suitable integral equation will be used to provide an impedance type relationship between the acoustic pressure and the normal particle velocity on the surface of the structure.
Siamak Amini, Paul John Harris, David T. Wilton

### 3. Numerical Solution of the Exterior Helmholtz Problem

Abstract
In this section some of the commonly used numerical methods for the solution of second kind integral equations of the form
$$[- \lambda \phi \left( p \right) + \int_s {k\left( {p,q} \right)} \phi \left( q \right)d{S_q} = f\left( p \right)\quad p \in S]$$
(3.1)
or in equivalent operator form
$$[\left( { - \lambda I + K} \right)\phi = f]$$
(3.2)
will be studied. The particular numerical scheme employed in this text is the collocation method, an example of a projection method. Collocation methods are undoubtedly the most widely used techniques in practical applications and will be discussed in detail later in this chapter.
Siamak Amini, Paul John Harris, David T. Wilton

### 4. The Dynamic Fluid-Structure Interaction Problem

Abstract
The previous chapters have considered the problem of determining the acoustic field around an arbitrary structure where the normal particle velocities on the surface were assumed known. The motion of the elastic structure will now be included in the analysis in order to predict the resulting sound radiation due to applied forces throughout the structure and also to model acoustic scattering by elastic structures taking the dynamic structural response into account. In this case it is necessary to solve simultaneously the Helmholtz equation in the fluid region D+ and the equations of motion of the structure in D-. This coupling is achieved by ensuring that at the surface S the normal particle velocity is continuous.
Siamak Amini, Paul John Harris, David T. Wilton

### 5. The Determination of the Response from Sonar Transducers

Abstract
In this chapter the method developed and analysed in the previous chapter is used to determine the sound field radiated by a piezoelectric ring sonar transducer. In particular the method will be used to determine the frequency which gives the peak response.
Siamak Amini, Paul John Harris, David T. Wilton

### Backmatter

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