Computation of an acoustic Transmissionline



This Wikibook explains, how the characteristics of an acoustic transmissionline can be computed. It is based on the appropriate German speaking Wikibook, which was widely translated with www.DeepL.com/Translator (free version).

Introduction
Transmissionline (transmission line) is the name given in loudspeaker acoustics to a channel, usually folded several times, through which the sound emitted to the rear by a loudspeaker passes. At the end of this channel is an opening through which the sound can exit. Under suitable conditions, this sound can be used to extend and improve the amplitude frequency response of an electrodynamic loudspeaker cabinet at the low-frequency end.

Based on idealised theoretical approaches to the radiation behaviour of loudspeakers and using the theory of quasi-homogeneous absorbers, this book deals with how the low-frequency range of a transmission-line loudspeaker cabinet can be dimensioned and acoustically damped. Because of the simple mathematical formalism, functions of complex numbers are used for this.

Loudspeaker radiation pattern


The amplitude frequency response $$A_0$$ of an electrodynamic loudspeaker with the total quality factor $$Q$$ and the natural frequency $$f_0$$ is calculated theoretically over the sound frequency $$f$$ as follows


 * $$A_0 (f) = \frac {\frac {f} {f_0}} {\sqrt {\frac {1} {Q^2} + \left( \frac {f} {f_0} - \frac {f_0} {f} \right)^2}}$$.

In the resonance case $$f = f_0$$ this equation simplifies to.


 * $$A_0 (f_0) = Q$$

With increasing frequency, however, the frequency response drops again, contrary to the statement of the equation above, because the loudspeaker membrane can no longer resonate due to its mechanical inertia. Further deviations occur in practice because natural frequencies other than $$f_0$$ also make small contributions to the frequency response. These natural frequencies are generally caused by unwanted vibration modes of the loudspeaker membrane.

Free speaker


If a loudspeaker is operated at the frequency $$f$$ without a baffle or enclosure, then at the point $$x = 0$$ (i.e. at the loudspeaker surface) over the time $$t$$ the sound wave emitted to the front interferes with the following


 * $$A_v = A(f,t) = A_0(f) \cdot e^{-i \omega t}$$

and the sound wave radiated to the rear


 * $$A_h = -A(f,t) = -A_0(f) \cdot e^{-i \omega t}$$

with the $$e$$, the  $$i = \sqrt {-1}$$ and the  $$\omega = 2 \pi f$$ destructive at all times:


 * $$ A_{tot} = A_v + A_h = 0$$.

This means that the total amplitude frequency response $$ A_{tot}$$ becomes zero as the two sound waves cancel each other out. This is also called.

Loudspeaker with baffle


The problems with the free loudspeaker can be circumvented if the loudspeaker is installed in a baffle that is as wide as possible in relation to the largest sound wavelength to be radiated. However, such a baffle usually has to have very large dimensions. At a minimum frequency of, for example, 30 hertz, the baffle would have to extend laterally by several metres.


 * $$ A_{tot} = A_v = A(f,t) = A_0(f)\cdot e^{-i \omega t}$$

where $$A_v$$ represents the forward radiated sound wave, which in this case is identical to the total radiated $$ A_{tot}$$.

Sound labyrinth


An alternative would be to install the loudspeaker in a so-called sound labyrinth. The sound energy of the sound wave emitted to the rear is destroyed here by multiple reflection and multiple scattering in combination with absorption. However, such a sound labyrinth must also be very large so that the sound can be absorbed to a large extent. As with the baffle, the disadvantage is that the sound energy radiated backwards by the loudspeaker is not used.

Closed loudspeaker cabinet


The ideal behaviour of a sound labyrinth is not achieved with closed cabinet loudspeakers. In addition to the sound wave $$A_v$$ radiated to the front, the sound waves $$A_h$$ radiated to the rear, which are reflected by the inner walls of the enclosure, can also escape through the speaker opening. As a result of the reflections, standing waves also occur, which have a negative effect on the amplitude and phase frequency response of the loudspeaker cabinet. Although this can be limited by filling the inside of the speaker cabinet with a sound-absorbing medium and by not aligning the cabinet walls parallel, this effect can never be completely avoided. Furthermore, especially at low frequencies, sound pressure can be so high that feedback occurs on the speaker cone, which, like the reflected sound waves, has a negative effect as a non-linear effect.

The total radiated sound wave $$ A_{tot}$$ results from the superposition of all sound waves:


 * $$ A_{tot} = A_v + B(A_h) + C(A_h)$$,

where the function $$B(A_h)$$ represents the sound waves damped and reflected in the loudspeaker cabinet and $$C(A_h)$$ the non-linear components due to feedback of the sound.

Partially ventilated loudspeaker cabinet


The high sound pressure is avoided in the so-called partially ventilated loudspeaker cabinets by a small opening that serves to equalise the pressure. If such loudspeaker cabinets are well damped and do not have parallel cabinet walls so that no standing waves can build up, they represent a quite good compromise, but the sound energy radiated to the rear is not used for sound reproduction here either.

Bass reflex loudspeaker cabinet


In bass-reflex loudspeaker cabinets, the natural frequency of the loudspeaker cabinet is deliberately used to amplify the sound radiation of the loudspeaker cabinet mostly in the low frequency range. The loudspeaker cabinet has an additional, usually forward-facing tube opening and thus acts as a Helmholtz resonator with the amplitude function $$B_v(f,t)$$. However, it is problematic to achieve a uniform frequency response in this way because there is usually too much amplification in a frequency band that is too narrow, and as a result the two sound waves radiated forwards from the loudspeaker $$A_v$$ and from the additional opening $$B_v$$ can be very poorly matched to each other. Even with bass-reflex loudspeaker cabinets, as with closed loudspeaker cabinets, standing waves can naturally occur in addition.

The total radiated sound wave $$ A_{tot}$$ thus essentially results from the superposition of two sound waves:


 * $$ A_{tot} = A_v + B_v$$,

where the function $$A_v$$ represents the sound wave radiated forward and $$B_v$$ represents the sound waves radiated out of the bass-reflex port.

Transmissionline without damping


The sound wave $$A_h$$ emitted backwards from the loudspeaker can be conducted through a sound channel with the length $$X_0$$, at the end of which there is an opening. This ideally resonance-free sound channel was named transmission line by Arthur R. Bailey, a Briton who taught at the University of Bradford. The sound wave $$A_{h,TL}$$ emerging from this opening can interfere with the sound wave $$A_v$$ radiated forward by the loudspeaker. In the case of the undamped sound channel, the amplitude of the interfered sound wave results from the sum of the two components:


 * $$A_v = A_0(f)$$ as in the case of the free loudspeaker

and


 * $$A_{h,TL} = -A_0(f)\cdot e^{-i \omega \cdot {X_0 \over c_0}}$$.

Here $$c_0$$ is the speed of sound in a gaseous sound medium (in air at 20° C and 60% relative humidity is $$c_0 = 344 {\frac {m} {s}}$$).

At the loudspeaker front, $$A_{tot,TL}$$ without damping thus results in:


 * $$A_{tot,TL} = A_v + A_{h,TL} = A_0(f)\cdot \left(1 - e^{-i \omega \cdot {\frac {X_0} {c_0}}}\right)$$

Qualitative description
In the 1960s, Bailey came up with the idea of allowing the sound energy radiated backwards by a woofer to "die" in a sufficiently long, damped sound channel that is open at the end, which he described in two articles in the trade magazine Wireless World. Damping is an integral part of Bailey's design.

The damping, the speed of sound and thus also the wavelength of the sound waves are frequency-dependent (dispersion) in such a low-pass filter. The lower the frequency or the longer the wavelength of the sound waves, the less they are damped. The higher frequency components are converted into thermal energy by the acoustic damping, and the very low frequency sound waves still emerging from the transmission line at the end can improve the amplitude frequency response at low frequencies if there is suitable interference with the direct, forward emitted sound waves of the woofer used.

The first commercial loudspeaker cabinets built according to this principle were first brought to market by the Cambridgeshire-based company Radford, which cooperated with Bailey, and later by IMF Electronics and in Germany by Lautsprecher Teufel, among others.

The sound wave emitted from the loudspeaker to the rear is increasingly attenuated in the sound channel, which is damped with long-fibre sheep's wool, for example, but very low frequencies can leave the sound channel only weakly attenuated and be used for sound reproduction. According to Bailey, the length of the duct should be at least about a quarter of the wavelength of the lower cut-off frequency, but it can also be longer. It can be bent or folded in various ways to fit in a conventional enclosure, but reflections (for example, with curved or bevelled surfaces) should be avoided.

Since this "non-resonant loudspeaker cabinet" (Bailey) destroys a large part of the sound energy radiated to the rear, the efficiency of transmission line loudspeakers is relatively poor, but thanks to powerful (transistor) amplifiers, this is usually not an issue. When correctly constructed, transmission line speakers are characterised by very good, sound-neutral (low) bass reproduction and very good impulse behaviour. However, due to size, weight and the relatively high manufacturing costs, they could not establish themselves on the market in the long run.

Quantitative calculation
In the following, the facts described by Bailey essentially only qualitatively are described quantitatively by means of corresponding acoustic formulae.

If the transmission line is damped with a porous, fibrous absorber, the sound wave amplitude $$A_{h,TL,D}$$ emerging from the transmission line can be approximated according to Fridolin Peter Mechel with the theory of quasihomogeneous absorbers:


 * $$A_{h,TL,D} = -A_0 (f)\cdot e^{-i \omega \cdot {X_0 \over c_0} \cdot \sqrt{1 - i \cdot {r \over {\rho_0 \cdot \omega}}}}$$

Where $$\rho_0$$ is the density of the gaseous sound medium (for dry air at 20° C is $$\rho_0 = 1.2 {\text{kg} \over \text{m}^3}$$) and $$r$$ is the specific flow resistance of the porous absorber material in Newton seconds per biquadrate metre. The greater the specific flow resistance, the greater the sound attenuation. It depends mainly on the material, the fibre fineness and the density $$\rho$$, i.e. the packing density, of the absorber and the degree of filling in the transmission line. Furthermore, it depends on the uniform distribution of the absorber, but also on the direction of the fibres and the sound frequency, so that only approximate guide values for some porous, fibrous absorbers are given in the following table:





For the entire transmission line loudspeaker cabinet, the following frequency-dependent amplitude results at the front of the loudspeaker cabinet with damping:


 * $$A_{tot,TL,D} = A_v + A_{h,TL,D} = A_0(f)\cdot \left(1 - e^{-i \omega \cdot {X_0 \over c_0} \cdot \sqrt{1 - i \cdot {r \over {\rho_0 \cdot \omega}}}}\right)$$

The amplitude curve results from the magnitude of the complex function $$A_{tot,TL,D}$$. This must be adjusted according to the parameters total quality of the woofer $$Q$$, length of the transmission line $$X_0$$ and specific flow resistance $$r$$ in such a way that the amplitude curve is as even as possible at the low frequencies. The two figures opposite are intended to illustrate the influence of the two parameters $$X_0$$ and $$r$$ for a given $$Q$$.

The parameters determined to be suitable can be implemented in the construction of a transmission line loudspeaker cabinet. However, the correct damping in particular must be checked experimentally and corrected if necessary. Furthermore, when driving the woofer and midrange driver, the amplitude resulting from the above equation must be


 * $$|A_{tot,TL,D}|$$

and phase difference


 * $$arg(A_{tot,TL,D})$$

at the crossover frequency between the system of transmission line and woofer on one side and midrange driver on the other side are balanced by means of the crossover.

Empirical values for the design of a transmission line loudspeaker


The cross-section of the transmission line directly behind the loudspeaker should be at least as large as the loudspeaker cone area or even slightly larger. Along the transmission line its cross-section can be reduced by a maximum of 25 percent.

The woofer should have a relatively high overall quality factor $$\left(Q \approx 1\right)$$.

The optimum value of the specific flow resistance $$r$$ must be checked and adjusted experimentally in the finished loudspeaker cabinet (for example with a frequency generator or with a noise generator and frequency analyser), as it is difficult to estimate for an absorber built into the transmission line.

The crossover is used to distribute the electrical signal to the woofers, midrange and, if necessary, the tweeters. To avoid undesirable acoustic effects, the lateral distances between midrange and woofer as well as between tweeter and midrange should be kept as small as possible. In addition, all speakers should be flush with the cabinet and lie in one plane.

The exit port of a transmission line can be considered constructively like a subwoofer.

Summary of the project

 * Target audience: This Wikibook is aimed at acoustically interested readers with mathematical training.
 * Learning objectives: This Wikibook describes a complete way to calculate a damped acoustic transmission line with complex-valued functions. Building on basic considerations of sound generation and radiation of loudspeakers, the functioning of such a transmission line is worked out through step-by-step extensions and additions.
 * Book sponsorship/contact person: User:Bautsch
 * Are co-authors currently desired? '''Yes, very much so. Corrections of obvious errors directly in the text; content please via discussion.
 * Guidelines for co-authors: Wikimedia-like.