Interference of sound waves in a moving fluid

We investigate sound propagation in a moving fluid confined in a randomly corrugated tube. For weak randomness and small fluid velocities $v^{\left(0\right)}$, the localization length $\xi$ shows extreme sensitivity to the variation of $v^{\left(0\right)}$. In the opposite limit of large fluid velocities, $\xi$ acquires a constant value which is independent of the frequency of the incident sound wave, the degree of randomness, and $v^{\left(0\right)}$ itself. Finally, we find that the variance ${\sigma }_{lnT}^2$ of the logarithm of transmittance $ln\left(T\right)$ is a universal function of the ensemble average $\langle lnT\rangle$, which is not affected by the fluid velocity.

Figure 1

Pressure profile $p^{\prime }\left(x\right)$ along a periodically modulated tube with corrugated areas characterized by $L_1=L_2,A_1=0.6A_2$ and a defect $L_d=0.05L_2,A_d=A_2$ in the middle of the tube (we consider $L_2=1,A_2=1$). In this figure, we report the first localized mode (at frequency $\omega =1.533[c_0/L_2]$). This mode is created due to interference effects between the periodic modulation and the defect. The cross section of the leads is $A_0=0.6A_2$ and the velocity of the fluid is $v^{\left(0\right)}=0.1c_0.$

Figure 2

Dimensionless localization length $\tilde{\xi }=\xi /(L_1^{\left(0\right)}+L_2^{\left(0\right)})$ vs the rescaled lead fluid velocity ${\beta }^{\left(0\right)}=v^{\left(0\right)}/c_0$. Various lines correspond to different disordered strengths $\delta$, mean corrugation width $L_1^{\left(0\right)},L_2^{\left(0\right)}$, and different frequencies $\omega$ [18]. The horizontal black dashed line is the result of RPA equation (12).

Figure 3

Variance of logarithm of transmittance ${\sigma }_{lnT}^2$ vs its mean value $\langle lnT\rangle$ for various disordered strengths ${\delta }_1,{\delta }_2$, mean widths $L_1^{\left(0\right)},L_2^{\left(0\right)}$, frequencies, and velocities $v^{\left(0\right)}$ in the interval $[0,0.8]$ [18]. Each parameter configuration is denoted with a different symbol. All data collapse to one universal curve characterizing also the $v^{\left(0\right)}=0$ case (black solid line).