# Linear and Second-Order Wave Theory

We saw in Conservation Laws and Boundary Conditions that the potential flow model for wave propagation is given Laplace's equation plus the free-surface conditions. In this section we present the linear and second order theory for these equations. The linear theory is valid for small wave heights and the second order theory is an improvement on this. However, neither of these theories work for very steep waves and of course the potential theory breaks down once the wave begins to break and completely different methods are required in this situation.

## Linearization of Free-surface Conditions

We use perturbation theory to expand the solution as follows

\begin{align} \zeta &= \zeta_1 + \zeta_2 + \zeta_3 + \cdots \\ \Phi &= \Phi_1 + \Phi_2 + \Phi_3 + \cdots \end{align}

where we are assuming that there exists a small parameter (the wave slope) and that with respect to this the $\Phi_i$ is proportional to $\epsilon^i$ or $O(\epsilon^i)$. We then derive the boundary value problem for $\zeta_i,\Phi_i$. Rarely we need to go beyond $i = 3$ (in fact it is unlikely that the terms beyond this will improve the accuracy.

In this section we will only derive the free-surface conditions up to second order. Remember that $\nabla^2 \Phi_i =0$ for all $i$ We expand the kinematic and dynamic free surface conditions about the $z=0$ plane and derive statements for the unknown pairs $(\Phi_1,\zeta_1$ and $(\Phi_2, \zeta_2)$ at $z=0$. The same technique can be used to linearize the body boundary condition at $U=0$ (zero speed) and $U\gt 0$ (forward speed).

## Kinematic condition

The fully non-linear kinematic condition was derived in Conservation Laws and Boundary Conditions and we begin with this equation

$\left ( \frac{\partial \zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right )_{z=\zeta} = \left ( \frac{\partial \Phi}{\partial z} \right )_{z=\zeta}$

We expand this equation about $\zeta = 0$, which we can do because we have assumed that the slope is small. In fact the slope is our parameter $\epsilon$. It is obvious at this point that the theory does not apply to very steep waves. This gives us the following equation

$\left( \frac{\partial\zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right)_{z=0} + \zeta \frac{\partial}{\partial z} \left( \frac{\partial\zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right)_{z=0} + \;\cdots = \left( \frac{\partial\Phi}{\partial z} \right)_{z=0} + \zeta \left( \frac{\partial^2 \Phi}{\partial z^2} \right)_{z=0} + \;\cdots$

where we have only taken the first order expansion. We then substitute our expressions

\begin{align} \zeta &= \zeta_1 + \zeta_2 + \cdots \\ \Phi &= \Phi_1 + \Phi_2 + \cdots \end{align}

and keep terms of $\ O(\varepsilon), \ O(\varepsilon^2)$, remembering that $\zeta_1\Phi_1$ is $O(\varepsilon^2)$ etc.

## Dynamic condition

The fully non-linear Dynamic condition was derived in Conservation Laws and Boundary Conditions and is given by

$\zeta (x,y,t) = -\frac{1}{g} \left( \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=\zeta}$
$\left . \begin{matrix} \zeta = \dfrac{1}{g} \left( \dfrac{\partial\Phi}{\partial t} + \dfrac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=0}\\ \dfrac{1}{g} \zeta \dfrac{\partial}{\partial z} \left( \dfrac{\partial\Phi}{\partial t} + \dfrac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=0} + \cdots \end{matrix} \right \} \begin{matrix} \zeta = \zeta_1 +\zeta_2 + \cdots \\ \Phi = \Phi_1 + \Phi_2 + \cdots \end{matrix}$

## Linear problem

The linear problem is the $O(\varepsilon)$ problem derived by equating the terms which are proportional to $\epsilon$.

This can be done straight forwardly and gives the following expressions

$\frac{\partial\zeta_1}{\partial t} = \frac{\partial\Phi_1}{\partial z} , \ z=0;$

which follows from the Kinematic equation and

$\zeta_1 = -\frac{1}{g} \frac{\partial\Phi_1}{\partial t}, \ z=0;$

which follows from the Dynamic equation. These are the linear free surface conditions.

### Derivation using Bernoulli's equation

The pressure from Bernoulli, $\omega$ constant terms set equal to zero, at a fixed point in the fluid domain at $\mathbf{x}=(x,y,z)$ is given by:

$P = - \rho \left( \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdots \nabla\Phi + gz \right);$

When then make the perturbation expansion for the potential and the pressure

$\Phi = \Phi_1 + \Phi_2 + \cdots$

and

$P = P_0 + P_1 + P_2 + \cdots$
.

This allows us to derive

$P_0 = -\rho g z \,,$

which is called the Hydrostatic pressure and

$P_1 = - \rho \frac{\partial\Phi_1}{\partial t}$

which is the linear pressure.

### Classical linear free surface condition

If we eliminate $\zeta_1$ from the kinematic and dynamic free surface conditions, we obtain the classical linear free surface condition:

$\begin{cases} \dfrac{\partial^2\Phi_1}{\partial t^2} + g \dfrac{\partial\Phi_1}{\partial z} = 0, \qquad z=0\\ \zeta_1 = - \dfrac{1}{g} \dfrac{\partial\Phi_1}{\partial t}, \qquad z=0 \end{cases}$

With:

$P_1 = - \rho \frac{\partial\Phi_1}{\partial t}, \qquad \mbox{at some fixed point} \ \mathbf{x}$

Note that on $z=0, \ P_1 \ne 0$ in fact it can obtained from the expressions above in the form

$P_1 = -\rho g \zeta_1, \qquad z=0$

So linear theory states that the linear perturbation pressure on the $z=0 \,$ plane due to a surface wave disturbance is equal to the positive (negative) "hydrostatic" pressure induced by the positive (negative) wave elevation $\zeta_1 \,$.

## Second-order problem

The second order equations can also be derived straight forwardly. The kinematic condition is

$\frac{\partial\zeta_2}{\partial t} + \nabla\Phi_1 \cdot \nabla\zeta_1 = \frac{\partial\Phi_2}{\partial z} + \zeta_1 \frac{\partial^2 \Phi_1}{\partial z^2}, \quad z=0$

and the dynamic condition

$\zeta_2 = - \frac{1}{g} \left( \frac{\partial\Phi_2}{\partial t} + \frac{1}{2} \nabla\Phi_1 \cdot \nabla\Phi_1 \right)_{z=0} - \frac{1}{g} \zeta_1 \frac{\partial^2\Phi_1}{\partial z \partial t}, \quad z=0$

Alternatively, the known linear terms may be moved in the right-hand side as forcing functions, leading to:

### Kinematic second-order condition

$\frac{\partial\zeta_2}{\partial t} - \frac{\partial\Phi_2}{\partial z} = \zeta_1 \frac{\partial^2 \Phi_1}{\partial z^2} - \nabla\Phi_1 \cdot \nabla\zeta_1; \quad z=0$

### Dynamic second-order condition

$\zeta_2 + \frac{1}{g} \frac{\partial\Phi_2}{\partial t} = - \frac{1}{g} \left( \frac{1}{2} \nabla\Phi_1 \cdot \nabla\Phi_1 + \zeta_1 \frac{\partial^2\Phi_1}{\partial z \partial t} \right)_{z=0}$

where the second order pressure is given by

$P_2 = -\rho \left( \frac{\partial\Phi_2}{\partial t} + \frac{1}{2} \nabla\Phi_1 \cdot \nabla\Phi_1 \right); \quad \mbox{at} \ \mathbf{x}$

The very attractive feature of second order surface wave theory is that it allows the prior solution of the linear problem which is often possible analytically and numerically. The linear solution is then used as a forcing function for the solution of the second order problem. This is often possible analytically and in most cases numerically in the absence or presence of bodies. Linear and second-order theories are also very appropriate to use for the modeling of surface waves as stochastic processes. Both theories are very useful in practice, particularly in connection with wave-body interactions.