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Acoustics - directivity

  1. Nov 26, 2006 #1
    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?
     
  2. jcsd
  3. Nov 26, 2006 #2

    turbo

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    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.
     
  4. Nov 26, 2006 #3

    FredGarvin

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    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.
     
  5. Nov 26, 2006 #4

    Clausius2

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

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

    where [tex]x=kasin\theta[/tex]

    Therefore, since [tex]k\sim 1/\lambda[/tex]

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

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

    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 [tex]a/\lambda>>1[/tex]

    I really think this makes sense physically speaking.
     
  6. Nov 26, 2006 #5
    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?
     
    Last edited: Nov 26, 2006
  7. Nov 27, 2006 #6

    FredGarvin

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    Like I mentioned, the k is called the "wave number" and is defined as
    [tex]k = \frac{\omega}{c} = \frac{2 \pi}{\lambda}[/tex] where [tex]\omega[/tex] is the frequency, [tex]c[/tex] is the speed of sound in the medium and [tex]\lambda[/tex] 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:

    [tex]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[/tex]

    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?
     
    Last edited: Nov 27, 2006
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