Sound waves frequency (Hankel functions)

[skrat]

Homework Statement

Let's study harmonic sound waves with frequency $\omega$, that is emitted by a long wire. Let's approximate the earth, above which the wire is, with an infinite rigid plate. If the space wasn't limited by the earth, than the velocity potential of the source would be $\Phi (\vec r, t)=-\frac{i\alpha }{4}H_0^{(1)}(k|\vec r - \vec d|)e^{-i\omega t}$, where vectors $\vec r$ and $\vec d$ are both in the crosswise plane (I imagine this the plane where it's normal vector and source wire are parallel) and where $\vec d$ describes the wire position.
What is the amplitude $\alpha$, if the radiated power per unit length is $p$. Density of the air is $\rho$ and sound velocity is $c$.

Homework Equations

The following expressions could help: $$H_0^{(1)}\asymp \sqrt{\frac{2}{\pi x}}e^{ix-i\pi /4}$$ and $$\int _0 ^{\pi }cos(xsin(y))dy=\pi J_0 (x)$$

The Attempt at a Solution

Hopefully, the first half of my solution is good, while I am not so sure about the second part.

So my $\Phi$ has to be $0$ and the Earth (at $y=0$). Let's use bit a different notation than the problem wants me to. $$\Phi (\vec r, t)_{\infty}\equiv \Phi (\vec r, t)=-\frac{i\alpha }{4}H_0^{(1)}(k|\vec r - \vec d|)e^{-i\omega t}$$ and now let $\Phi (\vec r, t)$ be the function I am looking for - the one, that has to be $0$ at the earth.

To find my $\Phi$, I shall say that it is a linear combination of $\Phi=\Phi _{\infty}\vec r,t) +g$, where $g(x,y,t)$ is unknown for now but makes sure that $\Phi$ is $0$ at the earth. This gives me a boundary condition $\Phi (x,y=0,t)=\Phi _{\infty}(x,y=0,t) +g=0$ which leaves me with $$g=-\Phi _{\infty}(\vec r,t)=\frac{i\alpha }{4}H_0^{(1)}(k|\vec r + \vec d|)e^{-i\omega t}$$ meaning $$\Phi (\vec r,t)=\frac{i\alpha}{4}(H_0^{(1)}(k|\vec r+\vec d|)-H_0^{(1)}(k|\vec r-\vec d|))$$ At least I hope so.
Now the second part. If I am not mistaken $\vec v=-\nabla \Phi$. The goal is to $P=\int \vec j d\vec S$ find $\vec j=\rho \vec v=-\rho \nabla \Phi$.

Now I am not sure:
1. If so far everything is ok
2. I have to write $\Phi$ in cartesian coordinates in order to get $\nabla \Phi$, don't I?

 

[BigJDubs]

3. Does the integral have to be done in polar coordinates (I think not)?4. If I should get back to the polar coordinates, is this the $\vec j$ i need?Thank you for your help!