Total velocity potential and velocity components (Wave equation Potentials)

[dustydude]

Homework Statement

hey I am having trouble figuring this out. The question states that i am to find the second component of velocity?

there is a plane wave incident on a surface at (0,0).

Homework Equations

the incident potential is
$$\Psi^{I} (x,t)= A\, e^{-i\omega (t- \frac{1}{c}(x\,sin\theta -y\,cos\theta)) }$$
the reflected potential is
$$\Psi^{R} (x,t)= A\, e^{-i\omega (t- \frac{1}{c}(x\,sin\theta +y\,cos\theta)) }$$

theta is the angle which the wave makes with the y axis.
i think there is an assumption that
$$\triangledown \times \mathbf{u} = 0$$ such that $$\mathbf{u}=\triangledown \Psi$$

The Attempt at a Solution

First is the total velocity potential then just,
$$\Psi= \Psi^{I}+ \Psi^{R}$$

and then the second component of u would be
$$\frac{\partial } {\partial y}(\Psi^{I}+\Psi^{R})$$

 

[lippyka]

= A\, e^{-i\omega (t- \frac{1}{c}(x\,sin\theta -y\,cos\theta)) }(-cos\theta) + A\, e^{-i\omega (t- \frac{1}{c}(x\,sin\theta +y\,cos\theta)) }(cos\theta)