Rossby waves

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11: 30.4 Functions of the First Kind

§30.4 Functions of the First Kind

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§30.4(i) Definitions

… ►If $\gamma =0$, ${\mathrm{𝖯𝗌}}_n^m(x,0)$ reduces to the Ferrers function ${𝖯}_n^m\left(x\right)$: … ►

§30.4(ii) Elementary Properties

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§30.4(iv) Orthogonality

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12: 13.28 Physical Applications

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§13.28(i) Exact Solutions of the Wave Equation

►The reduced wave equation ${\nabla }^{2}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi \eta }\cos \varphi$, $y=2\sqrt{\xi \eta }\sin \varphi$, $z=\xi -\eta$, can be solved via separation of variables $w=f_1(\xi )f_2(\eta ){\mathrm{e}}^{\mathrm{i}p\varphi }$, where …

13: 30 Spheroidal Wave Functions

Chapter 30 Spheroidal Wave Functions

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14: 36.13 Kelvin’s Ship-Wave Pattern

§36.13 Kelvin’s Ship-Wave Pattern

►A ship moving with constant speed $V$ on deep water generates a surface gravity wave. …Then with $g$ denoting the acceleration due to gravity, the wave height is approximately given by … ►

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Figure 36.13.1: Kelvin’s ship wave pattern, computed from the uniform asymptotic approximation (36.13.8), as a function of $x=\rho \cos \varphi$, $y=\rho \sin \varphi$. Magnify

►For further information see Lord Kelvin (1891, 1905) and Ursell (1960, 1994).

15: 30.7 Graphics

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§30.7(ii) Functions of the First Kind

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Figure 30.7.5: ${\mathrm{𝖯𝗌}}_n^0(x,4)$, $n=0,1,2,3$, $-1\le x\le 1$. Magnify

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Figure 30.7.6: ${\mathrm{𝖯𝗌}}_n^0(x,-4)$, $n=0,1,2,3$, $-1\le x\le 1$. Magnify

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Figure 30.7.7: ${\mathrm{𝖯𝗌}}_n^1(x,30)$, $n=1,2,3,4$, $-1\le x\le 1$. Magnify

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Figure 30.7.21: $|{\mathrm{𝑄𝑠}}_0^0(x+\mathrm{i}y,-4)|$, $-1.8\le x\le 1.8$, $-2\le y\le 2$. Magnify 3D Help

16: 30.18 Software

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SWF2: ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$.

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SWF3: ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$.

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SWF4: $S_n^{m(j)}(z,\gamma )$, $j=1,2$.

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§30.18(iii) Spheroidal Wave Functions

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17: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►The agreement of these solutions with two-dimensional surface water waves in shallow water was considered in Hammack et al. (1989, 1995).

18: 21.9 Integrable Equations

… ►Particularly important for the use of Riemann theta functions is the Kadomtsev–Petviashvili (KP) equation, which describes the propagation of two-dimensional, long-wave length surface waves in shallow water (Ablowitz and Segur (1981, Chapter 4)): …Here $x$ and $y$ are spatial variables, $t$ is time, and $u(x,y,t)$ is the elevation of the surface wave. … ►

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Figure 21.9.1: Two-dimensional periodic waves in a shallow water wave tank, taken from Hammack et al. (1995, p. 97) by permission of Cambridge University Press. … Magnify

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Figure 21.9.2: Contour plot of a two-phase solution of Equation (21.9.3). … Magnify

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19: 30.14 Wave Equation in Oblate Spheroidal Coordinates

§30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(i) Oblate Spheroidal Coordinates

… ►The wave equation (30.13.7), transformed to oblate spheroidal coordinates $(\xi ,\eta ,\varphi )$, admits solutions of the form (30.13.8), where $w_1$ satisfies the differential equation … ►

§30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids

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20: 30.15 Signal Analysis

§30.15 Signal Analysis

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§30.15(i) Scaled Spheroidal Wave Functions

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§30.15(ii) Integral Equation

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