Article title: Basic considerations on the practical method for predicting sound insulation performance of a single-leaf window

As a basic study of a practical method for predicting sound insulation performance of windows, this report presents a study of the sound reduction index of windows with single glazing below a critical frequency. First, results calculated by an existing theory for a single plate for the sound reduction indices are compared with measured results of actual windows to assess the theory’s applicability for evaluating the sound insulation performance of windows. Next, a regression analysis is employed to measured results of a certain number of actual windows to explore a further development of a more practical prediction. The following findings were obtained: (1) Sound reduction indices of actual fixed windows are predictable using Sewell’s transmission theory for a single plate. However, sound reduction indices of openable windows, especially those of sliding windows, are affected strongly by window frame gaps. Therefore, predicting sound reduction indices of all windows accurately is difficult if using only one theory. (2) The frequency slope of the window reduction index is much lower than that of the mass law. Regression analyses indicate that the frequency slope of the reduction index of all examined windows is 3.0 dB per octave, on average.


Introduction
Indoor environmental quality (IEQ) is of paramount importance in houses, offices, and most types of buildings. Among various environmental factors, quiet indoor acoustic environment is necessary both in living and working places for the quality of human activities within those spaces, and this depends on the sound insulation performance of exterior walls of buildings, e.g., [1]. The indoor acoustic environment depends mainly on the window sound insulation performance because windows are the weakest components of exterior walls in many cases [2]. For acoustic environment design, predicting the sound insulation performance of windows is crucially important. Many points of consideration exist in window design include obtaining natural ventilation, day lighting, and the view from buildings. Actually, these functions often share a trade-off relation with sound insulation, which has eventually led to development of a wide variety of window types, with sound insulation performance depending on the window type. This leads to another approach of the performance of windows, especially natural-ventilating windows, to evaluate them by perceptual factors using indoor soundscape approach [3]. However, still it is important to develop a practical prediction method for sound reduction index that is applicable for windows of different types.
Although the mass law of a single plate gives a slope of 6 dB per octave [1], reduction indices of single glazing in an actual window frame usually indicate different values. It is therefore considered that the mass law is inapplicable to an actual window of usual size. Some other effects that are not observed in walls are unique to actual windows, such as air gaps between sashes and frames or degrees of fixation depending on the window type. Additionally, the window pane size should be a consideration when predicting the sound insulation of windows. Therefore, a simple mass-law based discussion cannot be sufficient for the present purpose: we must start with a theory that can accommodate the panel size effect.
For this purpose, the theory developed by Sewell [4] is regarded as promising. Sewell developed an expression for reduction indices of finite single plates considering plate size effects, although the theory is limited to below coincidence frequency. Although Sewell's theory is not aimed at finite-sized sashed windows, it considers differences in radiation efficiencies depending on the plate size to include the plate size effect. Quirt studied sound transmission characteristics experimentally with modelled windows [5], and found good correlation with the experimental results. However his model was considerably simplified, it is still not very clear how well the theory can describe the sound insulation performance of actual windows. Regarding the gap, some cases exist in which effects are studied using numerical analysis [6]. However, no practical formula to predict or estimate window frame effects on the sound insulation performance of a window has been proposed. Iwase et al. [7][8][9][10] attempted to evaluate gap effects experimentally using modelled reverberation chamber.
However, they did not present definite results that are applicable to practical cases. As there are fewer studies which show the data of a certain number of actual windows, general characteristics of sound insulation performance of an actual window are less known. Therefore, establishing a practical method to predict the sound insulation performance of an actual window with a simple expression is of paramount importance.
The goal of this project is establishment of a practical method of the sound insulation performance of a window. As a basic study, a sound reduction index of a single-leaf window below its critical frequency is discussed herein. First, we applied Sewell's theory for a single plate by comparing the calculated and measured results of reduction indices of actual windows. Results of the comparison showed the applicability of Sewell's theory. Secondly, as another alternative for obtaining a practical solution, we tried to apply regression analyses to our measured data of reduction indices of windows. For this purpose, we used 83 set of measured data of windows of three types; regression lines for windows of each type were obtained to confirm the slope of the frequency characteristics of the reduction index. 4

Fundamental principles of theoretical prediction of reduction index
According to the well-known mass law, field-incidence-averaged sound reduction index R m is given by the following expression [11]. However, for small plates, it cannot agree with the measured results. In most cases, R m by Eq.
(1) does not agree with measured results of window's reduction index. Sewell derived a formula for evaluating a reduction index for finite plate with sound-induced vibration that is applicable below the coincidence frequencies as follows.
In those equations, k = ω/c expresses the wavenumber, F denotes the sample area, and Λ represents the ratio of shorter side length to longer side length of the sample.

Measurement
Measurements were taken according to JIS A 1416 [12], which is compatible with ISO 10140-2 [13]. All measurements were taken in reverberation chambers at the YKK AP central test centre, which has coupled reverberant rooms: the source room has 492.8 m 3 ; the receiving room has 264.5 m 3 volume. Reduction indices in the 1/3 octave bands were calculated from 100 to 5,000 Hz.

Comparing theoretical results and measurement results
The calculated results by Sewell's theory for the reduction index of a plate are compared with the measured results of windows of different types. Its applicability for predicting the sound insulation performance of windows is evaluated. The sketches of measured windows of three types (fixed, projected, sliding) are shown in Figure 1. All windows in this measurement consist of a glass and actual frames. The fixed window is not openable. The projected window and sliding window are defined in this report as follows. The projected window is an openable window that top and bottom of the frame are connected with the sash using friction stay, and the sliding window refers to a window composed of two sashes moving horizontally. The three windows are divided according to window glass thickness. The glazing considered here is commonly used float glass as a single glazing, with 5, 6, 8 mm thicknesses. Regarding the window size, the samples are organised to three categories according to the approximate sizes: window size F is within 15% of the represented value shown in figures. Up to three measured results are plotted in each graph. It is noteworthy that the data used for this comparison include particular trends of windows: thicker glazing tends to be used for higher-specification windows intended for use in high-rise buildings. Because these data are obtained from actual commercial windows, this tendency is reflected in this study. results are expected to vary by the type of window, because openable windows have more complex construction than fixed window. But according to Fig. 3, a major trend of the results of projected windows is similar to the trend found for fixed windows. The measured values agree with Sewell's calculated values rather than the mass law. As described for fixed window, the measured results are not considered to depend on the window size. However, at around 500-1,000 Hz, some downward deviation from Sewell's curve is apparent. This downward deviation can be attributed to the effects of gaps and degrees of fixation of the window frame.    Consequently, for prediction of the sound insulation performance of windows including sliding windows, Sewell's theory can be informative. However, it is impossible to accomplish by depending solely on Sewell's theory. It is necessary that the prediction method of reduction indices of windows be distinguished at least by the window type. Additionally, from the results in this section, shared trends are visible that reduction indices of windows have linear frequency characteristics with constant slopes in frequency range not affected by the coincidence effect. As a briefer method to express the characteristics of sound insulation of window, a regression analysis is performed in next section.

Analysis procedure
The statistical prediction method of the sound reduction index for single-glazed window is studied. A number of measured samples are divided based on the window type and glass thickness.
Because the window size effect is regarded as unimportant based on earlier discussion, this analysis is not divided by size. Windows are categorised into three types (fixed, projected, sliding), with glass of three thicknesses (5, 6, and 8 mm); up to 10 measured results of reduction indices falling into the categories are shown, respectively. The samples used for analyses are measured results of product windows. Linear regression is applied to the average of 10 samples in the range below half of critical frequency to observe the frequency characteristics without coincidence effect. This is because the coincidence effect appears above the half of the critical frequency. Some conditions did not get 10 samples, showing the number of samples in the graph. In addition, measured data of reduction indices of single glazing without window frames [14] and the field incidence mass law are shown in the same figure for reference. From the regression results, the slopes of reduction indices of windows of different types are calculated.

Analysis result
Analysis results are displayed in Fig. 6 for nine categories: from left to right, the results of 5, 6, Presumably, there are variable states from almost fixed windows to windows with a wide gap such as a sliding window. The average slope of all windows used in the present exam is 3.0 dB/ oct.
Furthermore, for regression, R 100 is presented as a cue for the height value. R 100 in Fig. 6 shows a value of the regression at 100 Hz that is readable from the graph. The thicker the glass which is used, the higher a value of R 100 is shown.

Application of regression
As ideas of the practical prediction method of the reduction index of a window, regression equations from the preceding section are explained. This regression application is also useful to confirm the slope of the frequency characteristics of the reduction index of windows. As an example, the case of a sliding window with 5-mm-thick glass is depicted in Fig. 7. The plots demonstrate other measured data of the window in the same category: The solid line shows the regression equation: The broken line shows the field incidence mass law for reference. The regression equation is a straight line that increases by 0.9 dB every 1/3 octave. Expressed as an equation, it becomes R = 9 log f + 3.1 [dB]. In this case, the regression line shows good agreement with measured data: the maximum error is -1.8 dB at 800 Hz. The measured value exceeds the regression line at middle frequencies, probably because this window has good airtightness. The value would be slightly lower than the regression line in these frequencies if a window with weak airtightness were measured. In any case, the measured values of the reduction indices of sliding windows would not deviate from the regression line. Consequently, this can be inferred as the simplest method of practical prediction of sound insulation performance for a window. This regression line is regarded as an important cue for practical prediction. However, the utility of the method is limited below the critical frequency.
Some study must be conducted of the prediction of the reduction indices of windows above the critical frequency.

Conclusions
A basic study was conducted to explore methods for practical prediction of the sound insulation performance of windows. First, Sewell's theory for walls, which incorporates panel size effects, was applied to windows. The results were compared with the measured results of window reduction indices. Next, a regression analysis was performed from the measured results of 10 samples for each window type and each glass thickness. Consequently, the following findings were obtained: 1. Sewell's theory showed good agreement with measured results obtained for fixed windows with single glazing. Sewell's theory for a plate appears to be applicable for predicting the reduction index of a fixed window. However, sound reduction indices of openable type windows, especially of sliding windows, were affected remarkably by window frame gaps. Consequently, predicting sound reduction indices of all windows accurately is difficult when using Sewell's formula alone.
Results also clarified that Sewell's formula overestimates the window pane size effect.
2. The frequency slope of windows is much lower than the general theory of mass law. There are characteristics of slopes below the critical frequencies for windows of different types. The frequency slope values for fixed windows are 3-3.3 dB/ oct. Those for sliding windows show 2.7-2.9 dB/ oct. Regression analysis shows that the reduction index slope with the frequency of all windows used is an average 3.0 dB/ oct.
As described in this report, important cues were obtained for practical prediction of the sound insulation performance of a window. However, the range of the problem remains limited to single