Wave scattering in spatially inhomogeneous currents

We analytically study a scattering of long linear surface waves on stationary currents in a duct (canal) of constant depth and variable width. It is assumed that the background velocity linearly increases or decreases with the longitudinal coordinate due to the gradual variation of duct width. Such a model admits an analytical solution of the problem in hand, and we calculate the scattering coefficients as functions of incident wave frequency for all possible cases of sub-, super-, and transcritical currents. For completeness we study both cocurrent and countercurrent wave propagation in accelerating and decelerating currents. The results obtained are analyzed in application to recent analog gravity experiments and shed light on the problem of hydrodynamic modeling of Hawking radiation.

Figure 1

Sketch of (a) accelerating and (b) decelerating background currents.

Figure 2

The sketch of a duct with the decreasing width that provides spatially accelerating background current.

Figure 3

The dispersion dependences for surface waves on uniformly moving shallow water. Lines 1 and 2 pertain to cocurrent and countercurrent propagating waves, respectively, in a subcritical current ($U<c_0$). Lines 3 and 4 pertain to positive- and negative-energy waves, respectively, in a supercritical current ($U>c_0$) both propagating downstream.

Figure 4

Modules of transformation coefficients as functions of dimensionless frequency $\widehat{\omega }$ for $V_1=0.1$, $V_2=0.9$. Line 1: $|T|$. Line 2: $|R|$. Line 3: $|B_1|$. Line 4: $|B_2|$. Dashed line 5 represents the asymptotic for the reflection coefficient $R\sim {\widehat{\omega }}^{-1}$.

Figure 5

Modules of function $\mathrm{\Phi }(\xi )$ for wave scattering in (a) accelerating and (b) decelerating subcritical currents with $V_1=0.1$ and $V_2=0.9$ in the former case and $V_1=0.9$ and $V_2=0.1$ in the latter case. Line 1 in each frame pertains to the cocurrent propagating incident wave, and line 2 to the countercurrent propagating incident wave. Dashed vertical lines show the boundaries ${\xi }_1$ and ${\xi }_2$ of the transient domain where the speed of the background current linearly changes.

Figure 6

Modules of transformation coefficients as functions of dimensionless frequency $\widehat{\omega }$ when (a) a positive-energy wave scatters and (b) a negative-energy wave scatters in the current with $V_1=1.1$, $V_2=1.9$. Line 1: $|T_p|$. Line 2: $|T_n|$. Line 3: $|B_1|$. Line 4: $|B_2|$. Dashed lines 5 represent the asymptotics for (a) $|T_n|\sim {\widehat{\omega }}^{-1}$ and for (b) $|T_p|\sim {\widehat{\omega }}^{-1}$.

Figure 7

Module of function $\mathrm{\Phi }(\xi )$ for the scattering of positive- and negative-energy waves (a) accelerating with $V_1=1.1$ and $V_2=1.9$ and (b) decelerating with $V_1=1.9$ and $V_2=1.1$ supercritical currents for two particular values of frequency, $\widehat{\omega }=1$ (line 1) and $\widehat{\omega }=100$ (line 2).

Figure 8

The dependences of energy transmission factors $K_{Tp}$ and $K_{Tn}$ on the frequency for a relatively small increase of current speed ($V_1=1.1$, $V_2=1.9$), lines 1 and 2, respectively, and a large increase of current speed ($V_1=1.1$, $V_2=8.0$), lines 3 and 4, respectively. Inclined dashed lines show the asymptotic dependences $K_{Tn}\sim {\widehat{\omega }}^{-2}$.

Figure 9

Modules of the transformation coefficients as functions of dimensionless frequency $\widehat{\omega }$ for $V_1=0.1$, $V_2=1.9$. Line 1: $|T_p|$. Line 2: $|T_n|$. Line 3: $|R|$. Line 4: $|B_2|=|{\breve{B}}_2|$. Dashed line 5 represents the asymptotic for $|T_n|\sim {\widehat{\omega }}^{-1}$.

Figure 10

The dependences of energy transmission factors on the frequency (i) when both $K_{Tp}<1$ (line 1) and $K_{Tn}<1$ (line 2) (here $V_1=0.1$, $V_2=1.9$); and (ii) when both $K_{Tp}>1$ (line 3) and $K_{Tn}>1$ (line 4) in a certain range of frequencies $\widehat{\omega }<{\widehat{\omega }}_c$ (here $V_1=0.9$, $V_2=8.0$). Inclined dashed lines show the asymptotic dependences $K_{Tn}\sim {\widehat{\omega }}^{-2}$.

Figure 11

Modules of function $\mathrm{\Phi }(\xi )$ for wave scattering in accelerating trans-critical current with $V_1=0.1$ and $V_2=1.9$ (line 1) and $V_1=0.9$ and $V_2=8.0$ (line 2). Dashed vertical lines 3 and 4 show the transition zone where the current accelerates from $V_1=0.1$ to $V_2=1.9$, and dashed vertical lines 5 and 6 show the transition zone where the current accelerates from $V_1=0.9$ to $V_2=8.0$. The plot was generated for $\widehat{\omega }=1$.

Figure 12

Modules of the transmission coefficients $|T_1|$ [line 1 in (a)] and $|T_2|$ [line 1 in (b)], as well as coefficients of wave excitation in the transient domain, $|B_1|$ (line 2), $|B_2|$ (line 3), $|{\widehat{B}}_1|$ (line 4), and $|{\widehat{B}}_2|$ (line 5), for the scattering of (a) positive-energy wave and (b) negative-energy wave as functions of dimensionless frequency $\widehat{\omega }$ for $V_1=1.9$, $V_2=0.1$. Dashed line 6 in (b) represents the high-frequency asymptotic for $|T_2|\sim {\widehat{\omega }}^{-1}$.

Figure 13

Modules of function $\mathrm{\Phi }(\xi )$ for wave scattering in decelerating transcritical current with $V_1=1.9$ and $V_2=0.1$ for $\widehat{\omega }=1$. Lines 1 and 2 pertain to the scattering of a positive-energy incident wave, and lines 1 and 3 pertain to the scattering of a negative-energy incident wave (line 1 is the same for both positive- and negative-energy waves).

Figure 14

Modulus of the reflection coefficients $|R|$ (line 1) and coefficients of wave excitation in the transient domain, $|B_1|$ (line 2) and $|B_2|$ (line 3), as functions of dimensionless frequency $\widehat{\omega }$ for $V_1=1.9$, $V_2=0.1$. Dashed line 4 represents the high-frequency asymptotic for $|R|\sim {\widehat{\omega }}^{-1}$.

Figure 15

Module of function $\mathrm{\Phi }(\xi )$ for a countercurrent propagating incident wave which scatters in the decelerating transcritical current with $V_1=1.9$ and $V_2=0.1$ for the particular values of $\widehat{\omega }$: line 1: $\widehat{\omega }=0.1$; line 2: $\widehat{\omega }=1$; and line 3: $\widehat{\omega }=100$.