https://www.win.tue.nl/~sjoerdr/papers/boek.pdf

An introduction to acoustics

We review the theory of wave propagation in one dimension through a medium consisting of N identical “cells.” Surprisingly, exact closed-form results can be obtained for arbitrary N. Examples include the vibration of weighted strings, the acoustics of corrugated tubes, the optics of photonic crystals, and, of course, electron wave functions in the quantum theory of solids. As N increases, the band structure characteristic of waves in infinite periodic media emerges.

Waves in locally periodic media

Fatigue testing at ultrasonic frequencies makes it possible to investigate the fatigue properties of materials in an effective and time saving manner. The mechanical and electrical components of ultrasound equipment are described in detail. Different testing and evaluation procedures are reviewed. Recommendations on how to perform high frequency fracture mechanics studies and to evaluate experimental results are described. Recent developments for multiaxial loading, for performing in-service loading fatigue experiments, and for realising high frequency torsional vibrations are described. A survey of fatigue investigations of metallic materials, composites, metallic glass, and ceramics, determined with the ultrasound method, is presented. Intrinsic frequency effects are reviewed, emphasising the influences on dislocation structures, crack initiation, and fatigue crack growth for pure metals and for technical materials. The advantages and disadvantages of the ultrasound method are compared with thos...

Fatigue crack growth and threshold measurements at very high frequencies

The propagation of acoustic waves in waveguides with variable cross section is considered using multimodal decomposition. The approach adopted is to construct two infinite first‐order differential equations for the components of the pressure and the velocity projected over the normal modes. From these an infinite matricial Riccati equation is derived for the impedance matrix. These equations are ordinary differential equations that can be integrated after truncation at a sufficient number of modes and take into account the coupling between modes. The stiffness of the pressure‐velocity equations induced by the presence of evanescent modes is avoided by first calculating the impedance matrix along the guide. The method is checked using different examples where the solutions of the plane‐wave approximation or the finite element method are known. Results show the method provides simple and accurate means to obtain the acoustic field with correct boundary conditions in a nonuniform guide with no restriction on the flare.

A study of wave propagation in varying cross-section waveguides by modal decomposition. Part I. Theory and validation

A variety of Industrial applications exist where power ultrasonic elements such as the ultrasonic horn are used. These included the Automotive, Instruments, Foods, Medical, Textiles and Material Joining and Fabrication Industries. In many of these devices the ultrasonic horn is the key component. The standard transducer used in these devices consists of three main parts, the backing, the piezoelectric elements and the horn. Standard horn designs have changed very little since their inception. There are four common types of standard horns. They are; constant, linear, exponential and stepped, which refer to the degree to which the area changes from the base to the tip. A magnification in the strain occurs in the horn that in general is a function of the ratio of diameters. In addition the device is generally driven at resonance to further amplify the strain. The resonance amplification is in general determined by the mechanical Q (attenuation) of the horn material and radiation damping. The horn length primarily determines the resonance frequency. For a 22 kHz resonance frequency a stepped horn of titanium has a length of approximately 8 cm. Although these standard horns are found in many current industrial designs they suffer from some key limitations. In many applications it would be useful to reduce the resonance frequency however this would require device lengths of the order of fractions of meters which may be impractical. In addition, manufacturing a horn requires the turning down of the stock material (eg. Titanium) from the larger outer diameter to the horn tip diameter, which is both time consuming and wasteful. In this paper we will present a variety of novel horn designs, which overcome some of the limitations discussed above. One particular design that has been found to overcome these limitations is the folded horn. In this design the horn elements are folded which reduce the overall length of the resonator (physical length) but maintain or increase the acoustic length. In addition initial experiments indicate that the tip displacement can be further adjusted by phasing the bending displacements and the extensional displacements. The experimental results for a variety of these and other novel horn designs will be presented and compared to the results predicted by theory.

Novel Horn Designs for Ultrasonic/Sonic Cleaning Welding, Soldering, Cutting and Drilling

Suppose that one solution of the “Webster” equation for any given horn is known, valid for particular boundary conditions and at one frequency. Expressions for the other linearly independent solution and for the derivatives of both solutions with respect to frequency are deduced and given in computable form. This allows the behavior of the horn for any loading, and in a wide range of frequencies, to be derived from a solution with one loading, at one frequency. The originally known solution may be computed, or experimental, or it may be the initial wavefunction in the inverse method of horn design (where the horn contour is derived from the initial wavefunction, which is constructed to satisfy the design requirements). With the new results, it is straightforward to compute the behavior of horns that were designed by the inverse method (for example, the “Terminated Gaussian” and the “Fourier”) in conditions not envisaged in their design. Further, the inverse method can now be used to design new horns with specified characteristics at a number of loads (including dissipative toads) and frequencies. The impedances of horns are discussed in the light of the relation between the solutions of Webster's equation, with special reference to the ideal transformer. The general impedance relations are thereby considerably simplified. Some basic results of the theory of horns are briefly reviewed and an annotated bibliography of over 200 references is given. Attention is drawn to very important work on the horn equation and its solution—long before Webster—in particular to brilliant papers by Daniel Bernoulli, by Lagrange, and by Euler in the 18th century.

Complete Solutions of the “Webster” Horn Equation

Using the design method described by Eisner and Seager, in which a suitable wavefunction is chosen and the corresponding profile is deduced, a new class of amplitude transformers is designed. The wavefunction used is a fourth‐order Fourier series. The resulting transformers give a useful compromise between the properties of the “stepped” and the “exponential” transformers. The output amplitude produced for a given maximum strain is somewhat lower than the amplitude given by the exponential transformer of the same magnification, but is much higher than that given by the corresponding stepped transformer. While the “Fourier” transformer is slightly less stiff in bending than the stepped, it is very much stiffer than the exponential.

Edward Eisner

Papers

Complete Solutions of the “Webster” Horn Equation

Design of Sonic Amplitude Transformers for High Magnification