Reflection of a Transient Acoustic Pulse by a Wall

Consider the reflection of a two-dimensional acoustic pulse by a plane wall as shown in Figure 6.13. The fluid is inviscid and is at rest at time t = 0. An acoustic pulse is generated by an initial pressure disturbance with a Gaussian spatial distribu­tion centered at (0, 20). The wall is located at y = 0. The initial conditions are as follows:

Подпись:– ln 2[x2 + (y – 20)2]

Подпись:u = v = 0.

In the numerical simulation, the 7-point DRP scheme is used, and the time step At is set equal to 0.07677. This value of At satisfies numerical stability requirement and ensures that the amount of numerical damping due to time discretization is insignificant.

Figure 6.13 shows the calculated pressure contour patterns associated with the acoustic pulse at 100, 300, and 500 time steps. The corresponding contours of the exact solution are also plotted in this figure. To the accuracy given by the thickness of the contour lines, the two sets of contours are almost indistinguishable. At 100 time steps, the pulse has not reached the wall, so the pressure contours are circular. At 300 time steps, the front part of the pulse reaches the wall. It is immediately reflected back. At 500 time steps, the entire pulse has effectively been reflected off the wall, creating a double-pulse pattern: one from the original source, and the other from the image source below the wall.

Figure 6.14 shows the computed pressure waveforms along the line x = y. The distance measured along this line from the origin is denoted by s. The computed waveforms at 400, 700, and 1000 time steps are shown together with the exact solution. As can be seen, there is excellent agreement between the exact and com­puted results. At 400 time steps, the pulse has just been reflected off the wall. At 700 time steps, the double-pulse characteristic waveform is fully formed. Both pulses propagate away from the wall with essentially the same waveform. The ampli­tude, however, decreases at a rate inversely proportional to the square root of the distance.