Modelling of a city canyon problem in a turbulent atmosphere using an equivalent sources approach
Introduction
Courtyards are shielded from direct traffic noise exposure by the surrounding buildings, and thereby they represent relatively quiet areas in urban environments. On a directly exposed façade, i.e., toward a street, the noise level can be sufficiently well predicted by standard methods based on ray-tracing (e.g., the Nordic calculation methods [1], [2]). Shielded areas seem more difficult to model. The sound paths contain multiple reflections involving diffraction, and the influence of streets further away is increased. A model for this kind of problem using equivalent sources has recently been developed for a homogeneous atmosphere [3]. Here, a further development is described, which incorporates effects of a turbulent atmosphere. The basis is a substitute sources method using a mutual coherence function for turbulence [4].
The situation of a depressed road, or a road surrounded by tall buildings, can be seen as a two-dimensional (2-D) problem, where the traffic will act as a line source and the road together with the buildings' façades will form a “city canyon”, the sending canyon. A shielded courtyard forms a second, receiving canyon.
The equivalent sources approach to the problem is field-based rather than ray-based, and thereby more easily captures the resonant behaviour of a city canyon. The original noise sources inside the sending canyon are exchanged for the equivalent sources at the top of the canyon. This can be seen as changing the position of a noise source from the canyon bottom to a typical roof height of the city, which also changes the strength and directivity of the source. The effect of turbulence is modelled on the equivalent sources on the canyon top, which is expected to be a more successful approach than using ray-based models including a scattering cross-section for turbulence. Such a scattering cross-section based method has been investigated previously and it was concluded that the turbulence influence increases at higher orders of the reflections inside the canyon [5]. This is because the higher order reflections correspond to ray directions that are more nearly horizontal, which makes the turbulence scattering stronger due to the smaller scattering angles. A precise calculation of the high order reflections together with turbulence scattering is difficult and the approach used here seems more promising.
Section snippets
A 2-D canyon solution using equivalent sources
In [6], the method of equivalent sources was used to calculate the insertion loss of balconies including absorbing surfaces. The main idea of the method is to reduce the problem to simplified geometries with boundary conditions which are easy to handle. On boundaries with different conditions, sources are placed. The strength of these sources are adjusted so that the boundary conditions are fulfilled everywhere. Applications of this can be found in [6], [7]. The method has been shown to be
Results
In the calculations, the canyons modelled are 18 m high and 19 or 11 m wide. All surfaces are acoustically hard. The source is on the bottom of the canyon, at position xs=9 or 5 m, for the 19 and 11 m wide canyon, respectively. (These data are summarised in Table 1.) For the single-canyon problems, the receiver is placed at x=500 m, on the hard surface. The results apply equally well to the reciprocal problem, with the receiver in the canyon and the source 500 m away. For the double-canyon
Conclusions
The increase in the sound pressure level due to turbulence can be predicted for city canyons. The model is based on the mutual coherence factor for sources or receivers separated in space in a turbulent atmosphere, and assumes a homogeneous and isotropic turbulence described by the von Kármán model.
The equivalent sources approach to the problem is expected to more easily capture the resonant behaviour of a city canyon than a ray-based model. The original noise sources inside the canyon can be
Acknowledgements
This paper is based on a study performed within the research programme “Soundscape Support to Health”, sponsored by the Swedish Foundation for Strategic Environmental Research (MISTRA), the Swedish Agency for Innovation Systems (Vinnova) and the Swedish National Road Administration (VV). The work behind this paper has also been funded by Formas (the Swedish Research Council for Environment, Agricultural Sciences and Spatial Planning).
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