John L. Davy

Predicting the Sound Insulation of Walls

Between 1990 and 1998, the author published five conference papers which described the gradual development of a simple theoretical model for predicting the sound insulation of building partitions. The first aim was to extend Sharps model for cavity walls to cavities without sound absorption. The second aim was to remove the reported over prediction of Sharps model for cavity walls. The third aim was to explain the five decibel empirical correction in Sharps model for stud walls. The fourth aim was to produce a more theoretically valid model than Gu and Wangs steel stud wall model. Although the simple theoretical model has been reasonably successful, several concerns have since arisen. This paper describes how these concerns have been addressed and gives the current version of this theoretical model for predicting sound insulation. The theoretical model is compared with a number of experimental measurements and produces reasonable agreement.

Predicting the sound insulation of single leaf walls: Extension of Cremer's model

In his 1942 paper on the sound insulation of single leaf walls, Cremer [(1942). Akust. Z. 7, 81-104] made a number of approximations in order to show the general trend of sound insulation above the critical frequency. Cremer realized that these approximations limited the application of his theory to frequencies greater than twice the critical frequency. This paper removes most of Cremers approximations so that the revised theory can be used down to the critical frequency. The revised theory is used as a correction to the diffuse field limp panel mass law below the critical frequency by setting the nonexistent coincidence angle to 90 degrees. The diffuse field limp panel mass law for a finite size wall is derived without recourse to a limiting angle by following the average diffuse field single sided radiation efficiency approach. The shear wave correction derived by Heckl and Donner [(1985). Rundfunktech Mitt. 29, 287-291] is applied to the revised theory in order to cover the case of thicker walls. The revised theory predicts the general trend of the experimental data, although the agreement is usually worse at low frequencies and depends on the value of damping loss factor used in the region of and above the critical frequency.

The forced radiation efficiency of finite size flat panels that are excited by incident sounda)

The radiation efficiency of an infinite flat panel that radiates a plane wave into a half space is equal to the inverse of the cosine of the angle between the direction of propagation of the plane wave and the normal to the panel. The fact that this radiation efficiency tends to infinity as the angle tends to 90 degrees causes problems with simple theories of sound insulation. Sato calculated numerical values of radiation efficiency for a finite size rectangular panel in an infinite baffle whose motion is forced by sound incident at an angle to the normal from the other side. This paper presents a simple two dimensional analytic strip theory, which agrees reasonably well with Satos numerical calculations for a rectangular panel. This leads to the conclusion that it is mainly the length of the panel in the direction of radiation, rather than its width that is important in determining its radiation efficiency. A low frequency correction is added to the analytic strip theory. The theory is analytically integrated over all angles of incidence, with the appropriate weighting function, to obtain the diffuse sound field forced radiation efficiency of a panel.

Sound transmission of cavity walls due to structure borne transmission via point and line connections

The author has published equations for predicting the air borne sound transmission of double leaf cavity walls due to the structure borne sound transmission across the air cavity via (possibly resilient) line connections, but has never published the full derivation of these equations. The author also derived equations for the case when the connections are rigid point connections but has never used them or published them or their derivations. This paper will present the full derivation of the authors theory of the air borne sound transmission of double leaf cavity walls due to the structure borne sound transmission across the air cavity via point or line connections which are modeled as four pole networks. The theoretical results will be compared with experimental results on wooden stud cavity walls from the National Research Council of Canada because the screw spacing is given for these results. This enables connections via studs and screws to be modeled as point connections and avoids the need to make any assumptions about the compliance of the equivalent point or line connections.

The prediction of flanking sound transmission below the critical frequency

Although reliable methods exist to predict the apparent sound reduction index of heavy, homogeneous isotopic building constructions, these methods are not appropriate for use with lightweight building constructions which typically have critical frequencies in or above the frequency range of interest. Three main methods have been proposed for extending the prediction of flanking sound transmission to frequencies below the critical frequency. The first method is the direct prediction which draws on a database of measurements of the flanking transmission of individual flanking paths. The second method would be a modification of the method in existing standards. This method requires the calculation of the resonant sound transmission factors. However, most of the approaches proposed to calculate the resonant sound transmission factor work only for the case of single leaf homogeneous isotropic building elements and therefore are not readily applicable to complex building elements. The third method is the measurement or prediction of the resonant radiation efficiency and the airborne diffuse field excited radiation efficiency which includes both the resonant and the non-resonant radiation efficiencies. The third method can currently deal with complex building elements if the radiation efficiencies can be measured or predicted. This paper examines these prediction methods.

The variance of the discrete frequency transmission function of a reverberant room.

This paper first shows experimentally that the distribution of modal spacings in a reverberation room is well modeled by the Rayleigh or Wigner distribution. Since the Rayleigh or Wigner distribution is a good approximation to the Gaussian orthogonal ensemble (GOE) distribution, this paper confirms the current wisdom that the GOE distribution is a good model for the distribution of modal spacings. Next this paper gives the technical arguments that the author used successfully to support the pragmatic arguments of Baade and the Air-conditioning and Refrigeration Institute of USA for retention of the pure tone qualification procedure and to modify a constant in the International Standard ISO 3741:1999(E) for measurement of sound power in a reverberation room.

The average specific forced radiation wave impedance of a finite rectangular panel.

The average specific forced radiation wave impedance of a finite rectangular panel is of importance for the prediction of both sound insulation and sound absorption. In 1982, Thomasson published numerical calculations of the average specific forced radiation wave impedance of a square of side length 2e for wave number k in half octave steps of ke from 0.25 to 64. Thomassons calculations were for the case when the forced bending wave number kb was less than or equal to k. Thomasson also published approximate formulas for values of ke above and below the published results. This paper combines Thomassons high and low frequency formulas and compares this combined formula with Thomassons numerical calculations. The real part of the approximate formula is between 0.7 dB higher and -1 dB lower than the numerical calculations. The imaginary part of the approximate formula is between 2.3 dB higher and -2.6 dB lower than the numerical calculations. This paper also gives approximate formulas for the case when kb is greater than or equal to k. The differences are between 0.8 and -1.2 dB for the imaginary part and between 6.2 and -2.4 dB for the real part.

The directivity of the sound radiation from panels and openings.

This paper presents a method for calculating the directivity of the radiation of sound from a panel or opening, whose vibration is forced by the incidence of sound from the other side. The directivity of the radiation depends on the angular distribution of the incident sound energy in the room or duct in whose wall or end the panel or opening occurs. The angular distribution of the incident sound energy is predicted using a model which depends on the sound absorption coefficient of the room or duct surfaces. If the sound source is situated in the room or duct, the sound absorption coefficient model is used in conjunction with a model for the directivity of the sound source. For angles of radiation approaching 90 degrees to the normal to the panel or opening, the effect of the diffraction by the panel or opening, or by the finite baffle in which the panel or opening is mounted, is included. A simple empirical model is developed to predict the diffraction of sound into the shadow zone when the angle of radiation is greater than 90 degrees to the normal to the panel or opening. The method is compared with published experimental results.

The directivity of the forced radiation of sound from panels and openings including the shadow zone

This paper presents a method for calculating the directivity of the radiation of sound from a two dimensional panel or opening, whose vibration is forced by the incidence of sound from the other side. The directivity of the radiation depends on the angular distribution of the incident sound energy. For panels or openings in the wall of a room, the angular distribution of the incident sound energy is predicted using a physical model which depends on the sound absorption coefficient of the room surfaces. For an opening at the end of a duct, the sound absorption coefficient model is used in conjunction with a two dimensional model for the directivity of the sound source in the duct. The finite size of the duct or panel is taken into account by using a two dimensional model for the real part of the radiation efficiency of the finite size panel or opening. For angles of radiation close to 90° to the normal to the panel or opening, the effect of the diffraction by the panel or opening, or by the finite baffle in which the panel or opening is mounted, needs to be included. The directivity of the radiation depends strongly on the length of the radiating object in the direction of the observer and only slightly on the width of the object at right angles to the direction of the observer. For panels, the plate wave impedance of the panel is used. Above its critical frequency, a single panel radiates strongly at the angle at which coincidence occurs. The method is compared with published experimental results.

An empirical model for the equivalent translational compliance of steel studs.

The effect of the resilience of the steel studs on the sound insulation of steel stud cavity walls can be modeled as an equivalent translational compliance in simple models for predicting the sound insulation of walls. Recent numerical calculations have shown that this equivalent translational compliance varies with frequency. This paper determines the values of the equivalent translational compliance of steel studs which make a simple sound insulation theory agree best with experimental sound insulation data for 126 steel stud cavity walls with gypsum plaster board on each side of the steel studs and sound absorbing material in the wall cavity. These values are approximately constant as a function of frequency up to 400 Hz. Above 400 Hz they decrease approximately as a non-integer power of the frequency. The equivalent translational compliance also depends on the mass per unit surface area of the cladding on each side of the steel studs and on the width of the steel studs. Above 400 Hz, this compliance also depends on the stud spacing. The best fit approximation is used with a simple sound insulation prediction model to predict the sound insulation of steel stud cavity walls whose sound insulation has been determined experimentally.