2 9/12 W

1. As noted in the previous homework, the velocity field of shallow water waves is near the surface given by
   
   $$u = \epsilon \sin(kx+\omega t) \qquad v = - \epsilon \cos(kx+\omega t)$$
   
   
   where amplitude $\epsilon$, wave number $k$, and frequency $\omega$ are all positive constants. Find and draw the streamlines of the flow. Do not assume epsilon is small in this case. Compare with the pathlines. Why are they not the same?

2. Draw the streakline coming from a generator at the origin, which is turned on at time $t=0$. Draw the streakline for times $\omega t = 0$, $\frac14\pi$, $\frac12\pi$, $\pi$, $2\pi$, $4\pi$, $6\pi$, .... In a separate graph, draw the particle path of the particle that was released at time zero. Compare its position at the given times with that in the streaklines. Hint: Read up on how to get streaklines in your notes.

3. Substitute the Eulerian velocity field of stagnation point flow into the (Eulerian) Euler equations. In the force per unit volume, include the gravity force per unit volume. Assume that gravity is in the minus $y$-direction. You get three equations for the pressure, one giving its $x$-derivative, one its $y$ derivative, and the third its $z$-derivative. More than one equation for a single scalar unknown $p$ is usually too much, but show that in this case, there is indeed a solution $p$ that satisfies all three equations. Find out what it is. Does it satisfy the Bernoulli law?

   Note: to find the pressure correctly, solve the Euler equation in the $x$-direction for the pressure. The integration constant will depend on $y$ and $z$. Substitute this result into the Euler equation in the $y$-direction to narrow down the integration constant. Then substitute this result into the Euler equation in the $z$-direction to narrow down the constant even more.