Solving Coupled PDEs with Forcing Function - Nick

[nickthequick]

Hi,

I am trying to simplify the following equations to get a relationship involving just $\eta$:

1) $\nabla^2 \phi(x,z,t) = 0$

for $x\in [-\infty,\infty]$ and $z\in [-\infty,0]$, $t \in [0,\infty]$

subject to the boundary conditions

2) $\phi_t+g \eta(x,t) = f(x,z,t)$ at z=0


3) $\eta_t = \phi_z$ at z=0


and

4) $\phi \to 0 \ as \ z \to -\infty$


Here, g is a constant, $\eta, \phi$ are the dependent variables of the system and f represents a forcing function. Another important constraint is that for systems I'm interested in, f is non zero only for a small time interval.

For the case where f=0, one can find that

$\eta_{tt}-\frac{g}{k} \eta_{xx} =0$

where k is the wavenumber of the system.

I want to find an analogous relation when forcing is present.

Any help is appreciated,

Nick

 

[kai_sikorski]

Seems to me like you need some conditions on the t=0 face of the domain.

 

[nickthequick]

In the case where $f=0$ it is clear, from the fact that $\eta(x,t)$ is governed by a wave equation, that we will need to know $\eta (x,0)$ and $\eta_t(x,0)$ to completely describe the (d'alambert) solution. Let us say that before the forcing occurs, we know both $\eta(x,0) , \eta_t(x,0)$.

I do not see how this helps me find the 'particular' solution to this system of equations.


This problem comes from physics - namely, it's the solution to (conservatively) forced, inviscid, irrotational surface gravity waves. The forcing that I'm interested acts in a 'spatially compact' region over a short time, say from $[t_o,t_o+\Delta t]$. As a first step, I'm trying to solve this in the limit that the forcing is all concentrated at a particular point in space and time $(x_o,z_o,t_o) =(0,0,0)$ but have not made any headway.

 

[nickthequick]

Also, an alternative way of looking at this problem is the following: The form of the Bernoulli equation in post 1 (condition 2) comes from

$\vec{u}_t=-\frac{1}{\rho} \nabla p + \vec{F}$

Where $\vec{F} = \vec{\nabla} f$.

The reason I took the route I did in post 1 was to avoid discussion of the pressure field, but an alternative way to look at this problem is by resolving this field. By taking the divergence of the Navier Stokes equation, we find

$\nabla^2 p = \nabla \cdot \vec{F}$

such that p=0 at z=0 and $\nabla p \to 0$ as $x \to \pm \infty$

If I can solve for the pressure field, then I can find the vertical velocity, $\phi_z$ at z=0 and then from there resolve the form of $\eta(x,t)$


I am trying to solve this for a very simple form of the forcing - namely $F = C_o \delta(x_o,z_,t_o) \ \hat{x}$ but have not made much progress.

 

[blue_raver22]

Hi Nick,

It seems like you are trying to solve a system of coupled partial differential equations (PDEs) with a forcing function. This can be a challenging problem, but there are several approaches you can take to simplify the equations and find a relationship involving just \eta.

One approach is to use separation of variables, where you assume that the solution can be written as a product of functions of different variables. In this case, you can write \phi(x,z,t) = X(x)Z(z)T(t) and substitute it into the first equation, which will result in three separate ordinary differential equations (ODEs) for each variable. You can then solve each ODE individually and combine the solutions to get a general solution for \phi. From there, you can use the boundary conditions to find a relationship between \eta and \phi that satisfies the given conditions.

Another approach is to use the method of characteristics, where you can transform the PDEs into a set of ODEs along characteristic curves. This can help you reduce the number of variables and simplify the equations. However, this method may not always be applicable to all types of PDEs.

In terms of finding an analogous relationship when the forcing function is present, you can try to incorporate the forcing term into your solution for \phi and see if you can still satisfy the boundary conditions. This may involve using techniques such as Green's functions or Fourier transforms. Additionally, you can also try to linearize the equations by assuming small perturbations and using perturbation methods to solve for the solution.

Overall, solving coupled PDEs with a forcing function can be a complex problem, but there are various techniques and methods that can help you simplify the equations and find a relationship involving just \eta. I hope this helps and good luck with your research!

Best,