Compare that to a more beaming (longer) horn:
- Those who like long horns simply don't like high frequencies
I'll just stick to what they say without supposing. As long as it is computed in the way the standard requires then it serves as a relative reference regardless of what either of us think.
From a physical point of view your graph doesn't looks correctly. Because it follows from your graph, that at lower frequency range where the real part of acoustic loading is lower, you get the higher values of acoustic power levels, and in the mid/hi range, where the real part acoustic acoustic loading is higher, you get the lower values of acoustic power level. I suppose you get this situation because the normalized SPL curves are used to estimate the "Sound Power" graph.
Say you calculate the total radiated power (i.e. approximate it based on whatever discrete measurement points and assumptions about the rest you have at hand). Now you would like to plot this power response into a SPL chart. So you convert this total power back into an average pressure, but what pressure is that and what is its meaning as a single curve? Try to think about that...
...and soon you'll find the answer:
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Ah Hah... Now I understand what you are getting at Mabat... The correct units for sound power is Watts. For convenience we want to express the sound power as Pascals, then convert to dB SPL. But a sound pressure level really only has meaning at a point in space. It is a pressure level along some axis, at some distance.
So for a device under test (DUT) that has a non-uniform radiation pattern (like a typical speaker), the sound power is the power (Watts) impinging upon an imaginary sphere surrounding the DUT. This value of power is converted into an equivalent dB SPL value, but this response curve does not represent any single point in space. No single measurement will reveal the power response curve, which means it is a hypothetical, or imaginary response curve. As a thought experiment, we can imagine a perfect omnidirectional source that has the same power response as the DUT. If we measure the pressure response of this device along any axis, we will measure the sound power response.
Do I understand your concept correctly?
j.
Yes, mathematically it is legal to apply EQ/to normalize, but perhaps it would be reasonable to point out that this is "Equalized Acoustic Power Response" (eg. "ESP") to distinguish the from the "Acoustic Power Response"("SP") under typical operation conditions, when the acoustic power droops in the vicinity of "the cutt-off frequency".
Why it shouldn't have a clear physical meaning? It has the same meaning as any other sound power response curve, no matter what EQ is used - as for a final loudspeaker that includes a crossover, which is a form of EQ. DI is not an "inverse of the equalized Sound Power" but a difference between a selected SPL curve and the SP curve, again no matter what is the EQ, if any.
Actually, I find it handy to show the SP curve as is. Call it whatever you like, it's the frequency response of an ideal omnidirectional source, radiating the same total power as the source with the FRs shown. - Sounds like a pretty clear physical meaning to me.
Actually, I find it handy to show the SP curve as is. Call it whatever you like, it's the frequency response of an ideal omnidirectional source, radiating the same total power as the source with the FRs shown. - Sounds like a pretty clear physical meaning to me.
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Because if you want to get an acoustic power response as it is described in physics (I'm physician, I'm not an engineer), you should integrate non-normalized acoustic pressure to get the correct quantity and dimention, otherwise you get some other quantity that you can name "equalized/normalized sound power/sound power level" or as it is conveniet to you, but clearly distiguish it from sound power that is normaly used in acoustics (I repeat once again that the peak of the radiated acoustic power of tht horn below it's "cut-off frequency" is a nonsense from a point of view of physics).
Below I wrote a few formulas, perhaps it will be easier to understand what I'm talking about.
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That's absurd in the same way as saying that a loudpeaker doesn't have a clearly defined power response because the raw responses of the individual drivers have been manipulated in the crossover. If you apply an EQ, you change the total radiated power (and the equivalent power response curve) in the same way as the frequency response. This has absolutely nothing to do with a cut-off or whatever.
A real system can have unusably low efficiency at low frequencies, but again, that doesn't imply anything regarding the power response curve, which just remains tied to its DI, that also doesn't depend of efficiency. In other words, you can describe a horn and its DI without even considering absolute efficiency, and still its power response curve will have a pretty well defined physical meaning, as I stated maybe five times already.
A real system can have unusably low efficiency at low frequencies, but again, that doesn't imply anything regarding the power response curve, which just remains tied to its DI, that also doesn't depend of efficiency. In other words, you can describe a horn and its DI without even considering absolute efficiency, and still its power response curve will have a pretty well defined physical meaning, as I stated maybe five times already.
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Given an acoustic power P watts radiating from a point source into a solid angle Ang steradians, the SPL in decibels at a distance of 1 metre from the point source is:
SPL = 20 * Log10(Pressure / Pref)
Where:
Pressure = Fractional space pressure magnitude = (P * rho * c / Area) ^ 0.5
Pref = Standard reference sound pressure of 20 micropascals = 20 * 10 ^ -6
rho = Density of air in kilograms per cubic metre = 1.205
c = Velocity of sound in air in metres per second = 344
Area = Fractional space SPL area for 1 metre radius = Ang
If P = 1 watt and Ang = 2.0 * Pi steradians (half space) then:
Pressure = (1 * 1.205 * 344 / (2.0 * Pi)) ^ 0.5 = 8.1224 pascals
SPL = 20 * Log10(8.1224 / (20 * 10 ^ -6)) = 112.1731 dB
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The directivity index DI is 10 times the logarithm to the base 10 of the directivity factor.
The directivity factor is the ratio of the intensity on a designated axis of a sound radiator at a given distance, to the intensity that would be produced at the same position by a point source if it were radiating the same total acoustic power as the radiator.
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Given a set of polars, in any absolute level or with an EQ applied (which is completely irrelevant), without any other knowledge of the underlying system, you can directly calculate the DI and the power response which that set represents. That's just a simple fact. And that power response is what it is - it's a frequency response of an ideal point source radiating the same total power as the system with the given polars.
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This is the procedure for axisymmetric sources as implemented and shown in the Ath reports for years.
For each frequency:
n = numer of polars
a[0 .. n-1] = polar angles
p[0 .. n-1] = pressure amplitudes
w[0 .. n-1] = summing weights
Power response as dB SPL:
For each frequency:
n = numer of polars
a[0 .. n-1] = polar angles
p[0 .. n-1] = pressure amplitudes
w[0 .. n-1] = summing weights
Power response as dB SPL:
Code:
// C++ code for the weights calculation
double cos_a0, cos_a1 = cos(a[0]);
const double cos_a2 = cos(a[n-1]);
const double St = 1.0 / (cos_a1 - cos_a2);
for(int i = 0; i < n - 1; ++i) {
cos_a0 = cos_a1;
cos_a1 = cos(0.5*(a[i] + a[i+1]));
w[i] = (cos_a0 - cos_a1)*St;
}
w[n - 1] = (cos_a1 - cos_a2)*St;