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?

 

[PJC01]

I can help clarify the equation and its significance in understanding air particle velocity. First, it is important to understand that the equation is describing a mathematical relationship between two physical quantities - air particle velocity and particle displacement. The cross product notation is commonly used in physics to show a relationship between two vectors, but it can also be used to show a relationship between two scalars.

In this case, the equation is showing that the air particle velocity (u) at a given time (t) is equal to the cross product of the angular frequency (ω) and the maximum particle displacement (x) at that same time. The angular frequency (ω) is related to the frequency (f) of the sound wave through the equation ω = 2πf, so it is also a scalar quantity.

To better understand this equation, let's break it down. The cross product of two vectors, A and B, is defined as A x B = |A||B|sinθ, where |A| and |B| are the magnitudes of the vectors and θ is the angle between them. However, since we are dealing with scalars in this equation, the magnitude of each scalar is simply its value, and there is no angle between them.

So, we can rewrite the equation as u(t) = ωx(t), where ω is the magnitude of the angular frequency and x(t) is the magnitude of the maximum particle displacement. This means that the air particle velocity (u) is directly proportional to the product of the angular frequency (ω) and maximum particle displacement (x). This makes sense because as the frequency of the sound wave increases, the air particles will be moving back and forth at a higher rate, resulting in a higher air particle velocity. Similarly, as the maximum particle displacement increases, the air particles will be moving over a larger distance, resulting in a higher air particle velocity.

In conclusion, the equation u(t) = jωx(t) is a mathematical representation of the relationship between air particle velocity and particle displacement. It may use cross product notation, but it is not a traditional cross product between two vectors. I hope this explanation helps you better understand the equation and its significance in the study of microphones and sound waves.