Special pattern in the directivity of loudspeaker

[hanson]

hi all.
For boxed loudspeaker, the direcitivity is as shown in the figure below.
The directivity change from a circular shape into a saw-tooth shape when the frequency increases. Why would this be?
I am wondering if this will be caused by interference? But there seem to be no evidence showing that interference will be stronger for higher frequencies?

Please express your views. Thanks

 

[FredGarvin]

The change in pattern is due to pressure nodes that are created at ever increasing frequencies.

The issue you talk about is a classical problem in acoustics; the baffled piston. The sound intensity can be calculated via

$$I_{(r,\theta)} = \frac{\rho_o c k^2 U_{rms} \pi^2 z^2}{4\pi^2r^2}\left[\frac{2J_1(k z sin \theta)}{k z sin\theta}\right]^2$$

Where:
$$I$$ = sound field intensity
$$\rho_o$$ = density of the medium
$$c$$ = speed of sound
$$k$$ = wavenumber
$$U_{rms}$$ = RMS speed of the piston surface
$$z$$ = piston radius
$$r$$ = radius from piston face
$$J_1$$ = first order Bessel function

The term in the second set of brackets is a directivity factor. That is the term that is responsible for the directional nature of the the pattern. Kind of interesting note: This is another reason why you see large woofers and small high frequency tweeters. The low frequency sounds are pretty much omni-directional. The higher frequencies require a smaller radius to be somewhat omni directional. It would not be good to have a speaker that has nodal points within your living room.

 

[hanson]

The change in pattern is due to pressure nodes that are created at ever increasing frequencies.

The issue you talk about is a classical problem in acoustics; the baffled piston. The sound intensity can be calculated via

$$I_{(r,\theta)} = \frac{\rho_o c k^2 U_{rms} \pi^2 z^2}{4\pi^2r^2}\left[\frac{2J_1(k z sin \theta)}{k z sin\theta}\right]^2$$

Where:
$$I$$ = sound field intensity
$$\rho_o$$ = density of the medium
$$c$$ = speed of sound
$$k$$ = wavenumber
$$U_{rms}$$ = RMS speed of the piston surface
$$z$$ = piston radius
$$r$$ = radius from piston face
$$J_1$$ = first order Bessel function

The term in the second set of brackets is a directivity factor. That is the term that is responsible for the directional nature of the the pattern. Kind of interesting note: This is another reason why you see large woofers and small high frequency tweeters. The low frequency sounds are pretty much omni-directional. The higher frequencies require a smaller radius to be somewhat omni directional. It would not be good to have a speaker that has nodal points within your living room.

Thanks FredGarvin.
But I don't quite understand why pressure nodes will be created when we increase the frequency. Can you explain in a bit more detail?

 

[FredGarvin]

You are looking at the result of really two waves interacting: the source wave from the speaker and the reflected wave from the, what would be referred to as, the ground or reflecting plane. As the frequency of the source increases, the wavelength goes down which means there will be more nodes within the same space to interact with each other. So, to get to point, it is due to interference between the two waves that causes the pattern.

 

[hanson]

Thanks FredGarvin again for the clear explanation.
You are so helpful.