# The Spec-Radiation Method for Layered Fluid Media
## 3 Results  
### Example with a Three-Layer Model  
A three-layer model was selected as an example in this section. This three-layer model is shown schematically in Figure 5. The excitation was an academic piston transducer with a diameter of \(\omega 40\) mm and an amplitude of one. The tilted orientation of \(5^{\circ}\) to the normal axis was computed with the procedure of Schmelt and Twiefel [47]. It emits pulse-like sinusoidal waves with a frequency of 50 kHz. The amplitude in the source plane and the time signal is displayed in Figure 6. The time step of the signal was \(\Delta t=2\times 10^{-7}\) s, and the signal was 0.000819 s long. Figure 5: Schematic representation of the 3-layer fluid model. Depicted are the incident \(E_{1}\), reflected \(R_{n,m}\), and transmitted \(T_{n,m}\) waves at fluid–fluid interfaces. \(n\) is the number of interfaces, and \(m\) is the number of interactions (example: \(R_{2,3}\) is the third reflection in the fluid at Interface 2). \(h_{n}\) is the height of the fluid layer; \(\rho_{n}\) is the density; \(\lambda_{n}\) is the wavelength. Figure 6: (**a**) Sound pressure amplitude of the tilted academic piston transducer in the plane \(z=0\) m; (**b**) transient pressure signal of the academic piston transducer. Fluids 1 and 3 were air with a speed of sound of 343 m/s and a density of 1.2041 kg/m\({}^{3}\). Fluid 2 was water with a speed of sound of 1484 m/s and a density of 997 kg/m\({}^{3}\). \(\Delta x=\Delta y=2\) mm; \(\Delta f_{x}=\frac{1}{x_{max}}\); \(\Delta f_{y}=\frac{1}{y_{max}}\). The simulation space had a size of 0.6 m \(\times\) 0.6 m \(\times\) 0.3 m (length \(\times\) width \(\times\) height). A step size in the \(z\)-direction of \(\Delta z=1\) mm was used in this example. The first interface was at \(h_{1}=0.1\) m, and the second interface was at \(h_{1}+h_{2}=0.2\) m. To calculate the sound pressure between the interfaces, the spec-radiation method without the extension as in Section 2.1 can be used. Here, only the reflected and transmitted parts, as depicted in Figure 5, were calculated. More reflections and transmission can be calculated, but to demonstrate how the procedure works, only these parts were determined. The individual layers between the interfaces are shown as cross-sections in Figure 7, as well as the superpositioned result of the half-3D volume. Here, each image was scaled in color to the maximum in the respective image to make the representation easier to understand. It can be seen that when all the results are superimposed, only with a color scaling down to \(-90\) dB related to the maximum of the total volume, after the second interface, the transmittance becomes visible. The influences of reflection and refraction are also clearly visible. In Medium 2, it can be seen from the much wider sound field that there was a much larger wavelength present. It can be seen that from the angle of incidence of 5\({}^{\circ}\), an angle of approximately 22\({}^{\circ}\) arose in Medium 2, as defined by [PERSON]'s law of refraction. Due to [PERSON]'s law, the refraction angle is here:  
\[\gamma_{2}=\arcsin\biggl{(}\frac{c_{2}}{c_{1}}\sin(5^{\circ})\biggr{)}\approx 2 2^{\circ}. \tag{23}\]  
Figure 7: Calculated sound fields in the layered fluids and the superposition of the sound fields. Figure 8 depicts the particle deflection at time \(t=0.4\) ms in a cross-section of the calculated volume. In Figure 8, a region between the plane \(z=0\) m and the second interface at \(z=0.2\) m is presented. It can be seen that at this time, the sound waves already reflected at the first interface. In Medium 2, the much longer wavelength, due to the higher speed of sound, was also evident. In Figure 8 is an enlarged section of the area depicted. The oscillation direction of the particles is displayed with vectors. The representation of the vectors was generated with the MATLAB function \"quiver\". The vectors clearly show that the waves were longitudinal, since the direction of oscillation was in the direction of propagation.