









 theory of sound  finite elements 


The finite element method has extensively been used in many engineering fields with great success. This method represents a general class of techniques for the approximate solution of partial differential equations whereby standard techniques such as Euler’s and RungeKutta method are used to integrate these equations. Apart from mechanical thermal and electrical problems also acoustic and even coupled acousticstructural problems can be handled with these methods. In the finite element formulation of acousticstructural problems, the necessary partial differential equations to describe the coupled problem can be derived from the conservation law of linear momentum. In what follows, some acousticstructural examples are presented to offer a first glimpse of the use of this method in sound problems. A first example is given in the experiment of Fig. 1 where a relatively large aluminum plate (large enough to make sure that no edge effects are involved) is used. This plate is immersed in water, which will act as the propagation medium of a 4MHz ultrasonic beam with a Gaussian width of 15 mm. The angle of the incident beam in Fig. 1 is slightly different from the critical Rayleigh angle. In this case we can see that the beam just reflects on the aluminum surface without splitting up. In a second experiment (Fig. 2), the beam radiates under an angle of 31° and the Schoch effect unfolds. This effect arises when an ultrasonic beam is incident at the Rayleigh angle. The reflected beam at the interface surface, in most cases, shows up as two lobes separated by a zone with no pressure distribution. The first lobe is generally called the specular lobe, the second one is usually referred to as the nonspecular lobe. The existence of the nonspecular lobe is a consequence of the generation of leaky Rayleigh waves.

FIGURE 1


Figure 2


A reproduction of this splitphenomenon of the ultrasonic beam is shown in what follows applying the finite element method. The outcome can be found in Fig. 3 and Fig. 4 where respectively a 25° and a 31° incident beam is modeled. In these figures the square of the pressure amplitude, which is proportional to the sound intensity, is shown which makes a comparison with the Schlieren pictures of Fig. 1 and Fig. 2 possible. In Fig. 3, the beam reflects on the aluminum surface in correspondence with the experiment shown in Fig. 1. In Fig. 4 it is clear that the beam splits in two lobes under the Rayleigh angle as anticipated.

Figure 3


Figure 4


A second example deals with the origins of the Schoch effect. Only recently conclusive experimental evidence was presented that the Schoch effect (with or without the generation of a null strip) is produced by the accompanying leaky Rayleigh waves. The experimental evidence is based on the use of a sound barrier positioned on a solid substrate immersed in water. By means of the finite element method a similar sound barrier was constructed on a solid plate making the study of the interaction of the bounded beam with the structure possible. The experimental acoustooptic Schlieren image is depicted in Fig. 5. Apart from the incoming sound beam on the left hand side, a complicated diffracted sound pattern is visible on the right hand side. Both sides are separated by a specially constructed sound barrier, consisting of two thin microscope slides. Using the finite element method, the sound barrier is constructed and shown in Fig. 6. It is clearly visible that the experimentally observed effects in Fig. 5 are completely retrieved.

Figure 5


Figure 6


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