## 3 Application of Integrated Transducer Arrays for Generation of Sound Waves
### Numerical Investigations  
A three-dimensional parametrical model consisting of two sub-models: a free-free supported plate and attached to it four piezoelectric transducers (transducer array), was elaborated (see Figure 4). The model has been created using commercially available finite-element software [27]. To realistically simulate the interaction between the electrical driving signal of the transducers and the resulting solid waves, a coupled electromechanical problem has to be solved. Therefore the transducers were modeled as 3-D, 20-node, coupled-field solid elements (SOLID 226). The elastic base structure has been meshed with 116,568 elements of type SOLID 186 with a maximum element size of 0.2 mm.  
In order to capture the time- and space-dependent phenomena connected with the propagation of mechanical waves, a transient analysis with the time step of 2.5 \(\upmu\)s has been performed. Herein, the termination time was set to 500 \(\upmu\)s which results in 200 load steps per one simulation. The mechanical properties of the base structure are presented in Table 1.  
Figure 4: General overview of the elaborated model with geometrical interpretation of the modifiable parameters.  
The transducers were driven by a Hann-windowed sine signal with 35 kHz center frequency and amplitude \(A\) equal to 6 V (Figure 5). The signal characteristics have been selected based on the technical specifications of the existing devices, planned to be used in the experimental investigations.  
In order to generate directional waves in the plate, the transducers have to be activated in predefined moments in order to amplify the desired plate mode as well as to shape the directional characteristics of the propagating wave. The geometrical configuration together with the material properties of the base structure governs the optimal time delay (\(\Delta t\)) which in the presented study equals 23.4 \(\upmu\)s (see Section 2).  
Since the analyzed phenomena are time- and location-dependent, a simple and informative presentation of the results is greatly limited. In order to visualize the wave field, snapshots at some time step have been created. For the results presentation, the wave field for one time step namely at 20 \(\upmu\)s has been selected based on the following criteria:  
* the wave has to propagate through some distance in order to reveal its directivity and plate mode character; and
* the reflections of higher Lamb modes--if any present--from the plate edges should not interfere with the main wave front;  
The obtained results (Figure 6) confirmed that through application deliberately-activated structure-integrated transducers organized into a linear array the generation of directional wave is possible. It is clearly visible that the wave amplitudes along the main wave propagation direction are amplified in comparison with those propagating in the other direction. It can be observed that some side lobes are also present. In order to confirm these observations experimental investigations on nominally-identical structures have been conducted.  
Figure 5: Transducer driving signals. (**A**) For the first transducer within the array. (**B**) For the second transducer within the array. (**C**) For the third transducer within the array. (**D**) For the fourth transducer within the array.  
Figure 6: Out-of-plane displacements of the plate wave generated by a transducer array at \(t=20\)\(\upmu\)s. For the animation of the wave field please refer to supplementary material.