Eigenfunction Matching for a Semi-Infinite Dock

Introduction

This is one of the simplest problem in eigenfunction matching. It also is an easy problem to understand the Wiener-Hopf and Residue Calculus

We begin with the simply problem when the waves are normally incident (so that the problem is truly two-dimensional. We then consider the case when the waves are incident at an angle. For the later we give the equations in slightly less detail.

Governing Equations

We begin with the Frequency Domain Problem for a dock which occupies the region [math]\displaystyle{ x\gt 0 }[/math]. The water is assumed to have constant finite depth [math]\displaystyle{ h }[/math] and the [math]\displaystyle{ z }[/math]-direction points vertically upward with the water surface at [math]\displaystyle{ z=0 }[/math] and the sea floor at [math]\displaystyle{ z=-h }[/math]. The boundary value problem can therefore be expressed as

[math]\displaystyle{ \Delta\phi=0, \,\, -h\lt z\lt 0, }[/math]

[math]\displaystyle{ \phi_{z}=0, \,\, z=-h, }[/math]

[math]\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0,\,x\lt 0, }[/math]

[math]\displaystyle{ \partial_z\phi=0, \,\, z=0,\,x\lt 0, }[/math]

We must also apply the Sommerfeld Radiation Condition as [math]\displaystyle{ |x|\rightarrow\infty }[/math]. This essentially implies that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave and a wave propagating away.

Solution Method

We use separation of variables in the two regions, [math]\displaystyle{ x\lt 0 }[/math] and [math]\displaystyle{ x\gt 0 }[/math].

Separation of variables

We now separate variables and write the potential as

[math]\displaystyle{ \phi(x,z)=\zeta(z)\rho(x) }[/math]

Applying Laplace's equation we obtain

[math]\displaystyle{ \zeta_{zz}+\mu^{2}\zeta=0. }[/math]

We then use the boundary condition at [math]\displaystyle{ z=-h }[/math], which is the same for all [math]\displaystyle{ x }[/math] to write

[math]\displaystyle{ \zeta=\cos\mu(z+h) }[/math]

where the separation constant [math]\displaystyle{ \mu^{2} }[/math] must satisfy different equations depending on whether [math]\displaystyle{ x\gt 0 }[/math] or [math]\displaystyle{ x\lt 0 }[/math]. For [math]\displaystyle{ x\lt 0 }[/math] the boundary condition

[math]\displaystyle{ k\tan\left( kh\right) =-\alpha,\quad x\lt 0\,\,\,(1) }[/math]

which is the Dispersion Relation for a Free Surface and for [math]\displaystyle{ x\gt 0 }[/math]

[math]\displaystyle{ \kappa\tan(\kappa h)=0,\quad x\gt 0 \,\,\,(2) }[/math]

Note that we have set [math]\displaystyle{ \mu=k }[/math] under the free surface and [math]\displaystyle{ \mu=\kappa }[/math] under the plate. We denote the positive imaginary solution of (1) by [math]\displaystyle{ k_{0} }[/math] and the positive real solutions by [math]\displaystyle{ k_{m} }[/math], [math]\displaystyle{ m\geq1 }[/math]. The solutions of (2) are [math]\displaystyle{ \kappa_{m}=m\pi/h }[/math], [math]\displaystyle{ m\geq 0 }[/math]. We define

[math]\displaystyle{ \phi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0 }[/math]

as the vertical eigenfunction of the potential in the open water region and

[math]\displaystyle{ \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad m\geq 0 }[/math]

as the vertical eigenfunction of the potential in the dock covered region. For later reference, we note that:

[math]\displaystyle{ \int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn} }[/math]

where

[math]\displaystyle{ A_{m}=\frac{1}{2}\left( \frac{\cos k_{m}h\sin k_{m}h+k_{m}h}{k_{m}\cos ^{2}k_{m}h}\right) }[/math]

and

[math]\displaystyle{ \int\nolimits_{-h}^{0}\phi_{n}(z)\psi_{m}(z) d z=B_{mn} }[/math]

where

[math]\displaystyle{ B_{mn}=\frac{k_{n}\sin k_{n}h\cos\kappa_{m}h-\kappa_{m}\cos k_{n}h\sin \kappa_{m}h}{\left( \cos k_{n}h\cos\kappa_{m}h\right) \left( k_{n} ^{2}-\kappa_{m}^{2}\right) } }[/math]

and

[math]\displaystyle{ \int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) d z=C_{m}\delta_{mn} }[/math]

where

[math]\displaystyle{ C_{m}=\frac{1}{2}h,\quad,m\neq 0 \quad \mathrm{and} \quad C_0 = h }[/math]

Therefore the potential can be expanded as

[math]\displaystyle{ \phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x\lt 0 }[/math]

and

[math]\displaystyle{ \phi(x,z)=\sum_{m=0}^{\infty}b_{m} e^{-\kappa_{m}x}\psi_{m}(z), \;\;x\gt 0 }[/math]

where [math]\displaystyle{ a_{m} }[/math] and [math]\displaystyle{ b_{m} }[/math] are the coefficients of the potential in the open water and the plate covered region respectively.

Incident potential

The incident potential is a wave of amplitude [math]\displaystyle{ A }[/math] in displacement travelling in the positive [math]\displaystyle{ x }[/math]-direction. The incident potential can therefore be written as

[math]\displaystyle{ \phi^{\mathrm{I}} =e^{-k_{0}x}\phi_{0}\left( z\right) }[/math]

An infinite dimensional system of equations

The potential and its derivative must be continuous across the transition from open water to the plate covered region. Therefore, the potentials and their derivatives at [math]\displaystyle{ x=0 }[/math] have to be equal. We obtain

[math]\displaystyle{ \phi_{0}\left( z\right) + \sum_{m=0}^{\infty} a_{m} \phi_{m}\left( z\right) =\sum_{m=0}^{\infty}b_{m}\psi_{m}(z) }[/math]

and

[math]\displaystyle{ -k_{0}\phi_{0}\left( z\right) +\sum _{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) =-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi _{m}(z) }[/math]

for each [math]\displaystyle{ n }[/math]. We solve these equations by multiplying both equations by [math]\displaystyle{ \phi_{l}(z) }[/math] and integrating from [math]\displaystyle{ -h }[/math] to [math]\displaystyle{ 0 }[/math] to obtain:

[math]\displaystyle{ A_{0}\delta_{0l}+a_{l}A_{l} =\sum_{n=0}^{\infty}b_{m}B_{ml}\,\,\,(3) }[/math]

and

[math]\displaystyle{ -k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l =-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(4) }[/math]

If we mutiple equation (3) by [math]\displaystyle{ -k_l }[/math] and add this to equation (4) we obtain

[math]\displaystyle{ -2k_{0}A_{0}\delta_{0l} =-\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} }[/math]

This equation gives the required equations to solve for the coefficients of the water velocity potential in the dock covered region.

Numerical Solution

To solve the system of equations (10) we set the upper limit of [math]\displaystyle{ l }[/math] to be [math]\displaystyle{ M }[/math]. We then simply need to solve the linear system of equations.

Solution with Waves Incident at an Angle

We can consider the problem when the waves are incident at an angle [math]\displaystyle{ \theta }[/math]. In this case we have the wavenumber in the [math]\displaystyle{ y }[/math] direction is [math]\displaystyle{ k_y = \sin\theta k_0 }[/math] where [math]\displaystyle{ k_0 }[/math] is as defined previously (note that [math]\displaystyle{ k_y }[/math] is imaginary).

This means that the potential is now of the form [math]\displaystyle{ \phi(x,y,z)=e^{k_y y}\phi(x,z) }[/math] so that when we separate variables we obtain

Therefore the potential can be expanded as

[math]\displaystyle{ \phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x\lt 0 }[/math]

and

[math]\displaystyle{ \phi(x,z)=\sum_{m=0}^{\infty}b_{m} e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x\gt 0 }[/math]

where [math]\displaystyle{ \hat{k}_{m} = \sqrt{k_m^2 + k_y^2} }[/math] and [math]\displaystyle{ \hat{\kappa}_{m} = \sqrt{\kappa_m^2 + k_y^2} }[/math] where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain

[math]\displaystyle{ A_{0}\delta_{0l}+a_{l}A_{l} =\sum_{n=0}^{\infty}b_{m}B_{ml} }[/math]

and

[math]\displaystyle{ -\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k]_{l}A_l =-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml} }[/math]

and these are solved exactly as before.