Plane waves satisfy Laplace's equation $\nabla^2\phi = 0$ for the velocity potential $\phi$. On the surface of the wave $y=h(x,t)$ the boundary conditions are given as $\frac{\partial \phi}{\partial t}+v_{1}\frac{\partial \phi}{\partial x} = v_{2}$ and $\frac{\partial \phi}{\partial t} + \frac{1}{2}v^2 = gh$, where $v_{1},v_{2}$ are the horizontal and vertical velocity components respectively. Transforming the variables $x,t,h,\phi$ into their non-dimensional forms $X,T,H,\Phi$ the equations may be expressed as $\nabla^2\Phi=0$ with the boundary conditions
$\frac{\partial H}{\partial T} + \epsilon\frac{\partial \Phi}{\partial X}\frac{\partial H}{\partial X}=\frac{\partial \Phi}{\partial Y}$ on $Y=\epsilon H(X,T)$
and
$\frac{\partial \Phi}{\partial T} + \frac{\epsilon}{2}((\frac{\partial \Phi}{\partial X})^2+(\frac{\partial \Phi}{\partial Y})^2)=\beta H$ on $Y=\epsilon H(X,T)$.
($\epsilon = \frac{h_{0}}{\lambda}$)
The first part of the question asks you to verify with the regular perturbation expansions that $\frac{\partial H_{0}}{\partial T} = \frac{\partial \Phi_{0}}{\partial Y}$, $\frac{\partial \Phi_{0}}{\partial T} = \beta H_{0}$ on $Y=0$. $\beta$ is a dimensionless constant known as the Froude number.
Question: After doing this, they ask you to write down the equations for the first-order perturbation for the wave height $H_{1}$. I don't quite understand exactly what they're asking me to do, are they asking me to find what equations the first-order terms satisfy? Are we looking for equations which explicitly contains $\epsilon$? I'm just not 100% what I'm supposed to do here.