Both you and your friend are partly right. (I was thinking as your friend just 20 min ago)
Let me rewrite <PERSON>'s radiating condition for the 2D case (just as an example, it works the same way in 3D).
$$\lim_{|\mathbf{x}|\to\infty}\sqrt{|\mathbf{x}|}\left(\frac{\partial u}{\partial|\mathbf{x}|}-iku\right)=0\quad\forall\theta$$
that is, the limit has to be $0$ when going to infinity in every direction.
As you say, if the function $u$ is smooth and goes to zero then the limit is fulfilled.
On the other hand your friend may have thought that the term between parentheses is null in the case of a plane wave. This is not completely true.
In 2D a plane wave takes the form: $$u\left(\mathbf{x}\right)=Ae^{ik\mathbf{d}\cdot\mathbf{x}}$$
being $A$ the amplitude of the wave, $k$ the wave number (the same as in the <PERSON> equation), $i$ the unit imaginary number, and $\mathbf{d}$ the unit vector in the direction of propagation of the wave, that is, perpendicular to the wavefront.
If we meassure our angles with respect to $\mathbf{d}$ then the plane wave can also be represented as:
$$u\left(\mathbf{x}\right)=Ae^{ik|\mathbf{x}|\cos(\theta)}$$
where we only evaluated the scalar product.
The derivative with respect to $|\mathbf{x}|$ is then:
$$\frac{\partial u}{\partial|\mathbf{x}|}=Aik\cos(\theta)e^{ik|\mathbf{x}|\cos(\theta)}=ik\cos(\theta)u\left(\mathbf{x}\right)$$
and we see that this only satisfies <PERSON>'s radiation condition when $\theta=0$.
So... 402
to sumarize:
Does a plane wave satisfy <PERSON>'s radiation condition?
No, it must be satisfied in all directions and it only satisfies it in the direction of propagation of the wave.
What does de $iku$ term means?
It is what you get when you differentiate a plane wave with respecto to $|\mathbf{x}|$ along its propagation direction.
What is the meaning of Sommerfield's radiation condition?
Sommerfeld's radiation conditions only allows waves to radiate energy towards infinity (outgoing waves) but not the infinity to radiate back. In the case of a plane wave the propagation of energy goes in only one direction in the whole plane, and with no attenuation (this is not very physical). It means that depending which direction are you looking at the energy is going towards infinity or coming from infinity.
The condition says that in every direction in space, the wave has to tend to a plane wave propagating in that direction, and the difference between the actual wave and a plane wave propagatin in that direction has to decrease faster than $\sqrt{|\mathbf{x}|}$.
To correct you, if a function $u$ decreases faster than $\sqrt{|\mathbf{x}|}$ in all directions, then this function will satisfy <PERSON>'s radiation condition, but this condition allow for more general cases.
Hope everything is clearer now. 346