# Hoffman's law

2354 Views 19 Replies 7 Participants Last post by  LTD02
somebody was kind enough to tell me what it is but now i want to know what is the most precise definition of it ?

if there is a definition with numbers in it i want that one. and preferrably with a link to where it was sourced from.

i want to ponder to exactly what extent it is true.
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are you serious?
proportional to box volume is easy enough to understand.

the intresting part is CUBE of cutoff frequency.

i assume by cube they mean 9db/oct ?

this is a very unusual number because a sealed box rolls off at 12db/oct.

do you have any thoughts on this ?
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think of how much more excursion is required to halve cutoff

Quote:
Originally Posted by noah katz /forum/post/16991569

think of how much more excursion is required to halve cutoff

i don't understand - please rephrase the question.

Quote:
Originally Posted by noah katz /forum/post/16991569

think of how much more excursion is required to halve cutoff
Quote:
Originally Posted by vasyachkin /forum/post/16991574

i don't understand - please rephrase the question.

4X displacement.

In some ways you can cheat and get efficiency by using a limited bandwidth solution like a TH, or alternatively you can just ignore the law to some extent by using ever increasing amounts of power and motor force to over come the very low efficiency to get some low bass out of a small air space. That has limits too though.

Does anyone have a link to a formula for what the theoretical maximum efficiencies for various sizes/frequencies would be?
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
Quote:
Originally Posted by Ricci /forum/post/16994322

4X displacement.

That's the main reason why my subwoofers are only good to near 18 Hz. Getting the FR flat down to 10 Hz requires 4X the displacement that I have in one form or another. I don't have that kind of real estate available!
4x displacement is 12db

where does 9db/oct come from ?

Quote:
Originally Posted by vasyachkin /forum/post/16996627

4x displacement is 12db

where does 9db/oct come from ?

With a Qtc of .707, excursion reaches a point and then stops with the same amount of power. That results in a 12db/oct drop.

Critically damped .5, drops off at ~9db/oct. Maybe that's where he came up with that number?
Could someone please show me an in-room FR of a sealed sub that reflects a 2nd order roll off from its anechoic corner?

No such thing.

Hoffman's law only exists in free space.

Bosso
I disagree. It has to do with efficiency at a certain frequency from a set enclosure volume. The environment that the system is in is acting as an EQ. Yes it can give you hellacious gain and thus effiicieny, but it is not actually increasing the total bass efficiency that is achievable from that set enclosure volume. It's 2 completely different things in my eyes.

Let's say that we have an airspace of 3 cu ft available for the bass system and 100w available and these values cannot change. There is only so much bass you can get out of that. You will never get 110db at 16hz out of it because you will never get a 90db efficiency at 16hz in 3 cu ft . If you had more airspace you could do a horn or an IB and get some decent output, or if you had 4000w you could get a little something going.
"Hoffman's law only exists in free space."

The room may change the predicted responses, but it ought to still determine the differences due to different box size, as 98% of the power at low freq is going to compress the air in it.
this 9db figure is either a mistake or something important that you don't understand

i have an idea for where it might come from but i really dont feel like thinking

to lower the corner frequency in the same box would require adding moving mass which would increase Q. to lower the Q back would require an increase in BL. this would partialy restore the efficiency.

NOW DO THE MATH ! I AM TOO OLD TO THINK !

well ... i could take a cup of coffee but who will pay me to do that ?
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
Quote:
Originally Posted by Ricci /forum/post/16998805

I disagree. It has to do with efficiency at a certain frequency from a set enclosure volume. The environment that the system is in is acting as an EQ. Yes it can give you hellacious gain and thus effiicieny, but it is not actually increasing the total bass efficiency that is achievable from that set enclosure volume. It's 2 completely different things in my eyes.

Let's say that we have an airspace of 3 cu ft available for the bass system and 100w available and these values cannot change. There is only so much bass you can get out of that. You will never get 110db at 16hz out of it because you will never get a 90db efficiency at 16hz in 3 cu ft . If you had more airspace you could do a horn or an IB and get some decent output, or if you had 4000w you could get a little something going.
Quote:
Originally Posted by noah katz /forum/post/17000207

The room may change the predicted responses, but it ought to still determine the differences due to different box size, as 98% of the power at low freq is going to compress the air in it.

Boundaries are not EQ. They boost efficiency by adding point sources, which increases the effective diaphragm area. The number of point sources increases as frequency decreases, proportional to room dimensions.

A larger box requires less amplification for frequencies below the corner, but decreases the number of point sources by lowering the frequency at which a point source is created, lowering efficiency above the corner.

The efficiency gained below the corner by using a large box is insignificant (approx 3dB, from 5Hz to the corner), but the loss of efficiency and potential above the corner (by use of the [let's change your example to] 500W amp and reduction in point source BW) is significant (from 500W to 4KW + smaller box in a corner = 10dB).

The loss in efficiency below the corner is swamped by the increase in efficiency by the boundaries in a corner load placement, which works with the gain in efficiency and potential above the corner from a smaller box and larger amplifier.

The above is based on use of a sealed subwoofer with no HP filter. Horns, Ported, PR'd, BP, or any other so-called higher efficiency alignment sub will lower the efficiency of the system due to the higher order roll off below the corner, size being irrelevant. They will also raise the overall non linear distortion of the system.

For example, a 20Hz tuned ported or horn sub adds 6-9dB of efficiency at 20Hz over a sealed sub (most of which is lost in-room to post EQ smoothing, unless you prefer a large hump at 20Hz in your FR), but loses 24-48dB of efficiency at 5Hz, while adding considerably to non linear distortion output from 5-20Hz.

Where's Hoffman's law during all of this? Way over there, quibbling over a dB or 2.

Bosso
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bosso,

Not sure I get what you're saying, but I don't see how the size of any of the boxes is relevant to whether it's a point source or not; assuming we're talking 30 Hz and below where wavelength is 37 ft.
ok here is the worked out math from the one and only mega genius - Vas !

take a sealed box system with Q = 1.0

now quadruple the MMS ->> corner frequency automatically decreases by 50% and output by 12 decibels

but thats not the whole story - the Q went from 1.0 to 2.0 ! ! !

we want to bring Q back to 1.0 so we're comparing apples to apples

we increase BL by a factor of sqrt(2) ...

efficiency across the board is increased by 3db while output at resonance damped by 3db and our Q returns to 1.0

there you have it - overall our corner frequency went down 1 octave and our system efficiency went down by 12 - 3 = 9 decibels ( 9 decibels = cube that Hoffman was talking about )

so in fact Hoffman's law proves the opposite of what you thought. it proves that efficiency is directly proportional to BL^2 ! ! !

another way of stating this is that for a given box size and corner frequency efficiency is inversely proportional to Q

3db corresponds to factor of 2 efficiency increase which was realized by increasing BL by a factor of sqrt(2) ...

so meditate on that my children ...

i skipped some math in my derivation if you don't understand any steps feel free to ask.
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
Quote:
Originally Posted by noah katz /forum/post/17001807

bosso,

Not sure I get what you're saying, but I don't see how the size of any of the boxes is relevant to whether it's a point source or not; assuming we're talking 30 Hz and below where wavelength is 37 ft.

Corner loading is 1/8 space. That would require the driver center to be no farther from all 3 boundaries than 28" for 60Hz, 21" for 80Hz, 17" for 100Hz, 11" for 160Hz.

Cancellations occur at 1/4 wavelength distances from any of those boundaries. At 200Hz, that would be 17", at 160Hz; 21", at 100Hz; 33", at 80Hz; 42"...

Complex variations of gain, no gain and cancellation can occur as box size increases and driver (or port) height from floor increases.

IOW, above the knee, specifically around the crossover, increasing box size limits the increase in efficiency from boundary gain, and can actually decrease efficiency, to a worse case scenario of necessitating the lowering of the crossover point.

Bosso
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"Cancellations occur at 1/4 wavelength distances from any of those boundaries."

Ah, so you're saying make it half of 1/4 WL; fair enough.

I was, assuming we're interested in efficiency in the deep bass, say 30 Hz and below, where your number would be 56".

And HIL is most relevant at the lowest freq.
perhaps we need a sticky on boundary cancellation, room gain, boundary gain, etc. this question keeps coming up and is key.
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