Onset patterns in a simple model of localized parametric forcing

We investigate pattern selection at onset in a parametrically and inhomogeneously forced partial differential equation obtained by generalizing Mathieu's equation to include spatial interactions. No separation of scales is assumed. The proposed model is directly relevant to the case of parametrically forced surface waves, such as cross-waves, excited by the horizontal vibration of a fluid, where the forcing is localized to a finite region near the endwall or wavemaker. The availability of analytical solutions in the limit of piecewise constant forcing allows us investigate in detail the dependence of selected eigenfunctions on spatial detuning, forcing width, damping, boundary conditions, and container size. A wide range of onset patterns are located and described, many of which are rotated, modulated, or both, and deviate far from simple crosswise oriented standing waves. The linear selection mechanisms governing this multiplicity of potential onset patterns are discussed.

Figure 1

Neutral stability curves as a function of $k_y$ with Neumann boundary conditions, $d=10\pi$, and $\gamma =0.01$. The dashed horizontal line marks the critical value in the uniformly forced case ($d=\infty$). Note that, despite near-tangencies that give the appearance of intersection on this scale, neutral stability curves are isolated and do not cross.

Figure 2

(a) Neutral stability curves as a function of $k_y$ with Neumann boundary conditions, $d=10\pi$, and $\gamma =0.01$. (b) Pattern corresponding to the minimum of the lowest tongue at $k_y=0.9988,f_c=0.02027$ for $t=0.9$ (chosen for aesthetic reasons). The dashed horizontal line in (a) marks the critical forcing in the uniform ($d=\infty$) case, while the vertical dashed line in (b) marks the edge of the forced region.

Figure 3

Modulation wavenumbers $k_1$ (solid curves), $k_2$ (dashed curves), and $k_3$ (dash-dotted curve) for the first four instability tongues as a function of $k_y$ with Neumann boundary conditions, $d=10\pi$, and $\gamma =0.01$. The splitting between $k_1$ and $k_2$ increases with successive modes (successively higher instability tongues).

Figure 4

The magnitude of the two solution components at the edge of the forced region, illustrated with the quantities $a_1\cosh \left({\kappa }_{1}d\right)$ and $a_2\cosh \left({\kappa }_{2}d\right)$, for Neumann boundary conditions, $d=10\pi$, and $\gamma =0.01$

Figure 5

Effect of detuning on pattern profile for the lowest neutral stability tongue. $\mathrm{Re}(P\left(x\right))$ is shown as a solid curve and the two components in the forced region, $a_j\cosh \left({\kappa }_{j}x\right)\cos \left({\varphi }_j\right),j=1,2$, as dash-dotted and dashed curves, respectively. Neumann boundary conditions are used with $d=10\pi$ and $\gamma =0.01$. The vertical dotted line marks the boundary of the forced region.

Figure 6

Effect of detuning on pattern profile for the second neutral stability tongue. $\mathrm{Re}(P\left(x\right))$ is shown as a solid curve and the two components in the forced region, $a_j\cosh \left({\kappa }_{j}x\right)\cos \left({\varphi }_j\right),j=1,2$, as dash-dotted and dashed curves, respectively. Neumann boundary conditions are used with $d=10\pi$ and $\gamma =0.01$. The vertical dotted line marks the boundary of the forced region.

Figure 7

(a) Neutral stability curves as a function of $k_y$ with Dirichlet boundary conditions, $d=10\pi$, and $\gamma =0.01$. (b) Pattern corresponding to the minimum of the lowest tongue at $k_y=0.9959$ and $f=0.02127$, for $t=1.1$. The dashed horizontal line in (a) marks the critical forcing in the uniform ($d=\infty$) case, while the vertical dashed line in (b) marks the edge of the forced region.

Figure 8

Modulation wavenumbers $k_1$ (solid curves), $k_2$ (dashed curves), and $k_3$ (dash-dotted curve) for the first four instability tongues as a function of $k_y$ with Dirichlet boundary conditions, $d=10\pi$, and $\gamma =0.01$. The splitting between $k_1$ and $k_2$ increases with successive modes (successively higher instability tongues).

Figure 9

Narrow forcing width $d=0.4\pi$ with Neumann boundary conditions and $\gamma =0.01$. (a) Lowest neutral stability curve as a function of $k_y$. (b) Pattern realized at the minima, $k_y=0.999$ and $f=0.1221$, for $t=1.1$. The dashed horizontal line in (a) marks the critical value in the uniformly forced case, while the vertical dashed line in (b) marks the edge of the forced region.

Figure 10

(a) Critical forcing curves as a function of $k_y$ with Dirichlet boundary conditions, $d=0.4\pi$, and $\gamma =0.01$. (b) Pattern corresponding to the minima of the instability tongue at $k_y=0.9327,f_c=1.6299$, and $t=0$ The dashed horizontal line in (a) marks the critical forcing in the uniformly forced case, while the vertical dashed line in (b) marks the edge of the forced region.

Figure 11

Dependence on forcing width with Neumann boundary conditions and $\gamma =0.01$. (a) Critical forcing value $f_c$ versus $\alpha =2\pi /d$. The dashed horizontal line shows the critical forcing in the uniformly forced case. (b) Wavenumbers $k_y$, $k_1$, and $k_3$; $k_2=0$ over this range. (c) ${\theta }_1=\arctan (k_y/k_1)$ and ${\theta }_3=\arctan (k_y/k_3)$.

Figure 12

Dependence on forcing width with Dirichlet boundary conditions and $\gamma =0.01$. (a) Critical forcing value $f_c$ versus $\alpha =2\pi /d$. (b) Wavenumbers $k_y$, $k_1$, $k_2$, and $k_3$. Note that $k_2$ (not labeled) is only nonzero for $\alpha <0.273$; this appears as a small bubble. (c) ${\theta }_1=\arctan (k_y/k_1)$, ${\theta }_2=\arctan (k_y/k_2)$ (not labeled), and ${\theta }_3=\arctan (k_y/k_3)$.

Figure 13

Dependence on damping with Neumann boundary conditions and $d=2\pi$. (a) Critical forcing value versus $\gamma$. The dashed curve shows the uniform forcing limit of Eq. (10). (b) Wavenumbers $k_y$, $k_1$, and $k_3$; $k_2$ is zero over this range of $\gamma$. (c) ${\theta }_1=\arctan (k_y/k_1)$ and ${\theta }_3=\arctan (k_y/k_3)$. Here $\log \left(\gamma \right)$ denotes the common logarithm (base 10).

Figure 14

Dependence on damping with Dirichlet boundary conditions and $d=2\pi$. (a) Critical forcing value versus $\gamma$. The dashed curve shows the uniform forcing limit of Eq. (10). (b) Wavenumbers $k_y$, $k_1$, $k_2$, and $k_3$. Note that $k_2$ is zero (i.e., ${\kappa }_2$ is real) for $\gamma >0.147$. (c) ${\theta }_1=\arctan (k_y/k_1)$, ${\theta }_2=\arctan (k_y/k_2)$, and ${\theta }_3=\arctan (k_y/k_3)$. Here $\log \left(\gamma \right)$ denotes the common logarithm (base 10).

Figure 15

Location of unmodulated subharmonic solutions with $d=0.4\pi$, $\gamma =0.01$, $L=15\pi$, and Neumann boundary conditions at $x=0,L$. The dotted curve shows the neutral stability curve in the semi-infinite case of Fig. 9. Most solutions for $k_y>1$ are not shown.

Figure 16

Location of unmodulated subharmonic solutions with $d=0.4\pi$, $\gamma =0.01$, $L=15\pi$, and Dirichlet boundary conditions at $x=0,L$. The dotted curve shows the neutral stability curve in the semi-infinite case of Fig. 10. Most solutions for $k_y>1$ are not shown.

Figure 17

Spacetime plots of two different eigenfunctions for the lowest solution point in Fig. 15 at $(k_y,f)=(0.993250.13101)$. The eigenfunction in (a) is concentrated near $x=0$ while that in (b) describes balanced excitation that is out-of-phase (by ${90}^{\circ }$) at each side. Dashed lines show the edge of the forced regions.

Figure 18

Projection of the neutral stability curves defined by Eq. (44) with $\gamma =0.02$, $d=2\pi /5$, and $L=30$ onto (a) the $(k_y,f)$ plane and (b) the $(k_y,\delta )$ plane. Neumann boundary conditions are applied at $x=0,L$. The dotted curve in (a) shows the neutral stability curve in the semi-infinite case, and the solid dots mark the location of unmodulated solutions ($\delta =0$). Solution branches with large $\delta$, including the one that approaches $(k_y,f)=(0,0)$ are associated with flat (long wavelength) solutions in Eq. (5).

Figure 19

Projection of the neutral stability curves defined by Eq. (44) with $\gamma =0.1$, $d=2\pi /5$, and $L=30$ onto (a) the $(k_y,f)$ plane and (b) the $(k_y,\delta )$ plane. Neumann boundary conditions are applied at $x=0,L$. The dotted curve in (a) shows the neutral stability curve in the semi-infinite case and the solid dots mark the location of unmodulated solutions ($\delta =0$).

Figure 20

Projection of the neutral stability curves defined by Eq. (44) with $\gamma =0.1$, $d=\pi$, and $L=30$ onto (a) the $(k_y,f)$ plane and (b) the $(k_y,\delta )$ plane. Neumann boundary conditions are applied at $x=0,L$. The dotted curves show the neutral stability curves in the semi-infinite case, and the solid dots mark the location of unmodulated solutions ($\delta =0$).