Understanding air particle velocity as cross product freq x disp

[The_Lobster]

I'm reading a book on microphones and came across the following:

The relation between air particle velocity (u) and particle displacement (x) is given by:
$u(t) = j\omega \times x(t)$

where $\omega = 2\pi f$ and $x(t)$ is the maximum particle displacement value.

and then it goes off talking about something else...

I feel stupid for asking this, but I don't get how the above equation works? For one, I thought cross products could only be be involving vectors? Aren't all the terms above scalars? Should I treat it as a dot product?

Any help in understanding the above, so I can see how the terms affect each other is greatly appreciated!

 

[AlephZero]

I don't think it is a cross product. It looks like the standard equation for velocity of a particle moving in simple harmonic motion.

I don't know why the book used $\times$ as a multiplication sign here.

 

[The_Lobster]

Thanks, AlephZero! Typical of me getting thrown off by poor notation...

 

[The_Lobster]

AlephZero: Are you saying that that equation is pretty much: $v = - A\omega \sin \omega t$? Does that mean I can consider the "maximum particle displacement" in the first equation, as the amplitude, A?