Why does the directivity of a loudspeaker change with frequency?

[hanson]

Hi all!
Can anyone explain me why the directivity of a loudspeaker varies with frequencies?
It is observed that the speaker is omni-directional at low frequencies and becomes increasingly forward-directional towards higher frequencies.
This is to due with the wavelength, but I have no clue on why it is so.

http://www.linkwitzlab.com/rooms.htm

This website mention some more observations but it doesn't provide explanaton.

Can anyone help explain me?

 

[turbo]

Google on loudspeaker directivity. There are good explanations on-line. Hint: the relation of the wavelength to the type and size of the enclosure is key.

 

[FredGarvin]

I remember studying monopoles and baffled pistons and calculating the pressure fields due to them in acoustics. Big fun. One thing that I remember is that the pressure distribution is dependent on a value known as "k*a" where k is the wave number and a is some characteristic dimension of the source. The wave number is a function of freqquency and is everywhere in acoustic theory. It's pretty cool to see how a piston's (monopole) pressure field changes with varying values of ka. Take a look here, most notably under section E:

http://www.gmi.edu/~drussell/GMI-Acoustics/Directivity-Frame.html

You'll get a feeling with how the directivity changes.

 

[Clausius2]

Nice link Fred. The only thing that is lacking there is this expansion of the Bessel function for large x:

$$J_1(x)\sim \sqrt\frac{2}{\pi x}cos (x+\pi/4)$$

where $$x=kasin\theta$$

Therefore, since $$k\sim 1/\lambda$$

-For $$a/\lambda<<1$$, that is for a baffle emitting with a long wave length compared with its size, the sound has no preferred direction to leading order.

-For $$a/\lambda>>1$$, the cos in the expansion of the Bessel function gets into the phase of the exponential:

$$P\sim F(r,x) e^{i(\omega t-kr+cos(a/\lambda sin \theta+\pi/4))}$$

showing the unidirectional character of the phase. That is, for large baffles compared with the wave length emitted, the sound is propagated with a preferred direction that coincides with the axis as $$a/\lambda>>1$$

I really think this makes sense physically speaking.

 

[hanson]

Thanks turbo, FredGarvin and Clausisus2.
wavelength and size of the enclosure?
With some search, I find the "ka" you guys mentioned.
But they are always expressed in terms of mathematical formulae that I have not yet learned before. I know the consequences of those equations but not the origin and derivation of them.

Could you offer a more physical explanation in plain terms?
Or how would you understand the phenomena without the equations?

 

[FredGarvin]

Like I mentioned, the k is called the "wave number" and is defined as
$$k = \frac{\omega}{c} = \frac{2 \pi}{\lambda}$$ where $$\omega$$ is the frequency, $$c$$ is the speed of sound in the medium and $$\lambda$$ is the wavelength.

In plain terms this is kind of tough. A lot of times in acoustics, things don't make sense until you go through the math. The only thing I can say is that if you look at the equation that describes the far field intensity of a piston (close to a speaker in mathematical models) you will see the following:

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

The second term in brackets is the directivity factor and you can see how k shows up in a lot of places. The a in this case is the piston's diameter. That directivity factor adjusts the pressure field intensity at different angles from the main axis of the piston.

Clear as mud, right?