Smooth Joint Connection of a Compression Driver to An OS-Horn

[whgeiger]

What follows is preliminary and probably has a mistake or two that I have not found yet. For the purposes of clarity, additional parentheses have been added to the formula. Any corrections needed are welcomed.

Regards,
WHG


Given:
[Rt] - Radius of Driver Exit & Horn Entry ...{06}
[At] - Flare Slope Angle at Driver Exit / Horn entry
= Atan([Mt]) ... [01]
[Mt] - Flair Slope at Driver Exit
= Tan[At] ... [02]
[Ac] - Horn Coverage Angle
= Atan([Mc]) ...[03]
[Mc] - Slope of Coverage Asymptote
= Tan[Ac] ...[04]
[L] - Length of Horn Profile

Find
[Lt] - Length of Horn Profile where its Slope Angle = [At] ...{15}
[Ro] - Radius of Horn Profile at [L] = 0 ...{18}

For
[Rl] - OS-Horn Radius at Profile Length [L]
= {([Mc]^2)*([L]^2)+([Ro]^2)}^(1/2) ...[05]

Characterize [Ro] in Terms of [Rt], [Lt] & [Mc]
[Rt] = {([Mc]^2)*([Lt]^2)+[Ro]^2}^(1/2) ...[06]
[Rt]^2 = ([Mc]^2)*([Lt]^2)+[Ro]^2 ...[07]
[Ro]^2 = -([Mc]^2)*([Lt]^2)+[Rt]^2 ...[08]

Characterize [Ro] in Terms of [Mt] & [Lt] & [Mc]
[Mt] = (([Mc]^2)*[Lt])/((([Mc]^2)*([Lt]^2)+([Ro]^2))^(1/2)) ...[09]
[Mt]^2 = (([Mc]^4)*[LT]^2)/(([Mc]^2)*([Lt]^2)+[Ro]^2) ...[10]
([MC]^2)*([LT]^2)+([Ro]^2) = (([Mc]^4)/([Mt]^2))*([LT]^2) ...[11]
([Ro]^2) = ((([Mc]^4)/([Mt]^2))-([MC]^2))*([LT]^2) ...[12]

From [08] & [12] Solve for [Lt]
([Mc]^4)/([Mt]^2))*([Lt]^2)=[RT]^2 ...[13]
[Lt]^2 = ([Rt]^2)*([Mt]^2)/([Mc]^4) ...[14]
[Lt] = {([Rt]^2)*([Mt]^2)/([Mc]^4)}^(1/2) ...[15]

From [08] & [14] Solve for [Ro]
[Ro]^2 = -([Mc]^2)*([Rt]^2)*([Mt]^2)/([Mc]^4)+([Rt]^2) ...[16]
[Ro]^2 = ([Rt]^2)*(1-([Mt]^2)/([Mc]^2)) ...[17]
[Ro] = [Rt]*{(1-([Mt]^2))/([Mc]^2)}^(1/2) ...[18]

[whgeiger]

Smooth Joint Connection of an OS-Horn to a Compression Driver

What follows is preliminary and probably has a mistake or two that I have not found yet. For the purposes of clarity, additional parentheses have been added to the formula. Any corrections needed are welcomed.

Regards,

WHG/09-Oct-10

*Update/Corrections
WHG/09-Oct-10

Given:
[Rt] - Radius of Driver Exit & Horn Entry ...{06}
[At] - Flare Slope Angle at Driver Exit / Horn entry
= Atan([Mt]) ... [01]
[Mt] - Flair Slope at Driver Exit
= Tan[At] ... [02]
[Ac] - Horn Coverage Angle
= Atan([Mc]) ...[03]
[Mc] - Slope of Coverage Asymptote
= Tan[Ac] ...[04]
[L] - Length of Horn Profile

Find
[Lt] - Length of Horn Profile where its Slope Angle = [At] ...{15}
[Ro] - Radius of Horn Profile at [L] = 0 ...{18}

For
[Rl] - OS-Horn Radius at Profile Length [L]
= {([Mc]^2)*([L]^2)+([Ro]^2)}^(1/2) ...[05]

Characterize [Ro] in Terms of [Rt], [Lt] & [Mc]
(Recast [05].)*
[Rt] = {([Mc]^2)*([Lt]^2)+[Ro]^2}^(1/2) ...[06]
[Rt]^2 = ([Mc]^2)*([Lt]^2)+[Ro]^2 ...[07]
[Ro]^2 = -([Mc]^2)*([Lt]^2)+[Rt]^2 ...[08]

Characterize [Ro] in Terms of [Mt] & [Lt] & [Mc]
(1st. Derivative of [06].)*
[Mt] = (([Mc]^2)*[Lt])/((([Mc]^2)*([Lt]^2)+([Ro]^2))^(1/2)) ...[09]
[Mt]^2 = (([Mc]^4)*[LT]^2)/(([Mc]^2)*([Lt]^2)+[Ro]^2) ...[10]
([MC]^2)*([LT]^2)+([Ro]^2) = (([Mc]^4)/([Mt]^2))*([LT]^2) ...[11]
([Ro]^2) = ((([Mc]^4)/([Mt]^2))-([MC]^2))*([LT]^2) ...[12]

From [08] & [12] Solve for [Lt]
([Mc]^4)/([Mt]^2))*([Lt]^2)=[RT]^2 ...[13]
[Lt]^2 = ([Rt]^2)*([Mt]^2)/([Mc]^4) ...[14]
[Lt] = [Rt]*[Mt]/([Mc]^2) ...[15]*

From [08] & [14] Solve for [Ro]
[Ro]^2 = -([Mc]^2)*([Rt]^2)*([Mt]^2)/([Mc]^4)+([Rt]^2) ...[16]
[Ro]^2 = ([Rt]^2)*(1-([Mt]^2)/([Mc]^2)) ...[17]
[Ro] = [Rt]*{1-([Mt]^2)/([Mc]^2)}^(1/2) ...[18]*

[whgeiger]

Originally Posted by whgeiger

Smooth Joint Connection of an OS-Horn to a Compression Driver

What follows is preliminary and probably has a mistake or two that I have not found yet. For the purposes of clarity, additional parentheses have been added to the formula. Any corrections needed are welcomed.

Regards,

WHG/09-Oct-10

*Update/Corrections
WHG/09-Oct-10

Given:
[Rt] - Radius of Driver Exit & Horn Entry ...{06}
[At] - Flare Slope Angle at Driver Exit / Horn entry
= Atan([Mt]) ... [01]
[Mt] - Flair Slope at Driver Exit
= Tan[At] ... [02]
[Ac] - Horn Coverage Angle
= Atan([Mc]) ...[03]
[Mc] - Slope of Coverage Asymptote
= Tan[Ac] ...[04]
[L] - Length of Horn Profile

Find
[Lt] - Length of Horn Profile where its Slope Angle = [At] ...{15}
[Ro] - Radius of Horn Profile at [L] = 0 ...{18}

For
[Rl] - OS-Horn Radius at Profile Length [L]
= {([Mc]^2)*([L]^2)+([Ro]^2)}^(1/2) ...[05]

Characterize [Ro] in Terms of [Rt], [Lt] & [Mc]
(Recast [05].)*
[Rt] = {([Mc]^2)*([Lt]^2)+[Ro]^2}^(1/2) ...[06]
[Rt]^2 = ([Mc]^2)*([Lt]^2)+[Ro]^2 ...[07]
[Ro]^2 = -([Mc]^2)*([Lt]^2)+[Rt]^2 ...[08]

Characterize [Ro] in Terms of [Mt] & [Lt] & [Mc]
(1st. Derivative of [06].)*
[Mt] = (([Mc]^2)*[Lt])/((([Mc]^2)*([Lt]^2)+([Ro]^2))^(1/2)) ...[09]
[Mt]^2 = (([Mc]^4)*[LT]^2)/(([Mc]^2)*([Lt]^2)+[Ro]^2) ...[10]
([MC]^2)*([LT]^2)+([Ro]^2) = (([Mc]^4)/([Mt]^2))*([LT]^2) ...[11]
([Ro]^2) = ((([Mc]^4)/([Mt]^2))-([MC]^2))*([LT]^2) ...[12]

From [08] & [12] Solve for [Lt]
([Mc]^4)/([Mt]^2))*([Lt]^2)=[RT]^2 ...[13]
[Lt]^2 = ([Rt]^2)*([Mt]^2)/([Mc]^4) ...[14]
[Lt] = [Rt]*[Mt]/([Mc]^2) ...[15]*

From [08] & [14] Solve for [Ro]
[Ro]^2 = -([Mc]^2)*([Rt]^2)*([Mt]^2)/([Mc]^4)+([Rt]^2) ...[16]
[Ro]^2 = ([Rt]^2)*(1-([Mt]^2)/([Mc]^2)) ...[17]
[Ro] = [Rt]*{1-([Mt]^2)/([Mc]^2)}^(1/2) ...[18]*

[Ed LaFontaine]

There is an excel worksheet available for this.

You have pm requesting your address.

[whgeiger]

Originally Posted by Ed LaFontaine

There is an excel worksheet available for this.

You have pm requesting your address.

Hi Ed,

I have an Excel sheet as well that was published on the web about 10-Years ago.
Just adding Freehafer's horn to it. So published the work here for others to use.
Like most other horns, it needs a Tractrix bell termination to make it finite.
A smooth match can be implemented as well by applying the same math regimen presented here.
Check your private e-mail for my address.

Regards,
Bill

[Ed LaFontaine]

Bill, ygm

[Ed LaFontaine]

Bill,

Any comments?

[whgeiger]

[QUOTE=Ed LaFontaine;2337948]Bill,

Hi Ed,

In your spredsheet, calculate [Lt] and [Ro] from [Rt] and [Mt] (or [At]) using the formula [15] and [18] provided here earlier.
Then recast formula [05] to
[Rl] - OS-Horn Radius at Profile Length [L]
= {([Mc]^2)*(([Lt]+[L])^2)+([Ro]^2)}^(1/2) ...[19]

Now the horn graph will start at the desired attachment point of [L]=0,[Rl]=[Rt] .
I will have an update shortly that will add a tractrix bell and refine the formula given here. These you may incorporate into your spreadsheet as well. I prefer horns with elliptical, rectangular, or 'bow tie' apertures, rather than those with circular boundaries that extend beyond the throat. This preference is based on the shape of the space being irradiated (typically of rectangular section) and the distribution of listeners to be covered within. Ideally the OS horn flair should be incorporated into the phase plug design of its accompanying compression driver. Presently most available drivers employ an exponential flair; so at best, the resulting combination yields a hybrid horn.

Regards,
Bill

[whgeiger]

To terminate Freehafer's OS Horn with a Tractrix Bell,

Let

[Ln] - Length of OS Horn Neck

Then

[Rn] - Radius of OS Neck at [Ln]
= SqRt(([Mc]^2)*([Ln]^2)+([Ro]^2)) ...[20]

[Mn] - Slope of OS Neck at Point [Ln],[Rn]
= ([Mc]^2)*[Ln]/SqRt(([Mc]^2)*([Ln]^2)+([Ro]^2)) ...[21]

[Mb] - Slope of Tractrix Bell at Point [Ln],[Rn] ...[22]
= [Rn]/SqRt(([Rm]^2)-([Rn]^2))

[Rm] - Radius of Tractrix Bell Mouth
= SqRt(([Mn]^2)+1)*[Rn]/[Mn] ...[23]

[Lb] - Length of Tractrix Bell
= [Rm]*LN(([Rm]+SqRt(([Rm]^2)-([Rn]^2)))/[Rn])-SqRt(([Rm]^2)-([Rn]^2)) ...[24]

Comments, corrections, and additions by others are welcome.

Regards,

WHG

[454Casull]

Originally Posted by whgeiger

Ideally the OS horn flair should be incorporated into the phase plug design of its accompanying compression driver. Presently most available drivers employ an exponential flair; so at best, the resulting combination yields a hybrid horn.

Regards,
Bill

For some drivers it should be possible to machine a new throat of simple cylindrical cross section and have the OS flare solely in the waveguide section. I wonder how this would affect the polar response and HOMs.