The effect of pasture utilization rate on stocks of soil organic carbon and total nitrogen in a semi-arid tropical grassland

The influence of grazing management on total soil organic carbon (SOC) and soil total nitrogen (TN) in tropical grasslands is an issue of considerable ecological and economic interest. Here we have used linear mixed models to investigate the effect of grazing management on stocks of SOC and TN in the top 0.5 m of the soil profile. The study site was a long-term pasture utilization experiment, 26 years after the experiment was established for sheep grazing on native Mitchell grass (Astrebla spp.) pasture in northern Australia. The pasture utilization rates were between 0% (exclosure) and 80%, assessed visually. We found that a significant amount of TN had been lost from the top 0.1 m of the soil profile as a result of grazing, with 80% pasture utilization resulting in a loss of 84 kg ha−1 over the 26-year period. There was no significant effect of pasture utilization rate on TN when greater soil depths were considered. There was no significant effect of pasture utilization rate on stocks of SOC and soil particulate organic carbon (POC), or the C:N ratio at any depth; however, visual trends in the data suggested some agreement with the literature, whereby increased grazing pressure appeared to: (i) decrease SOC and POC stocks; and, (ii) increase the C:N ratio. Overall, the statistical power of the study was limited, and future research would benefit from a more comprehensive sampling scheme. Previous studies at the site have found that a pasture utilization rate of 30% is sustainable for grazing production on Mitchell grass; however, given our results, we conclude that N inputs (possibly through management of native N2-fixing pasture legumes) should be made for long-term maintenance of soil health, and pasture productivity, within this ecosystem.

Contested Lands: Land Disputes in Semi-Arid Parts of Northern Tanzania : Case Studies of the Loliondo and Sale Divisions

This study Contested Lands: Land disputes in semi-arid parts of northern Tanzania. Case Studies of the Loliondo and Sale Division in the Ngorongoro District concentrates on describing the specific land disputes which took place in the 1990s in the Loliondo and Sale Divisions of the Ngorongoro District in northern Tanzania. The study shows the territorial and historical transformation of territories and property and their relation to the land disputes of the 1990s'. It was assumed that land disputes have been firstly linked to changing spatiality due to the zoning policies of the State territoriality and, secondly, they can be related to the State control of property where the ownership of land property has been redefined through statutory laws. In the analysis of the land disputes issues such as use of territoriality, boundary construction and property claims, in geographical space, are highlighted. Generally, from the 1980s onwards, increases in human population within both Divisions have put pressure on land/resources. This has led to the increased control of land/resource, to the construction of boundaries and finally to formalized land rights on village lands of the Loliondo Division. The land disputes have thus been linked to the use of legal power and to the re-creation of the boundary (informal or formal) either by the Maasai or the Sonjo on the Loliondo and Sale village lands. In Loliondo Division land disputes have been resource-based and related to multiple allocations of land or game resource concessions. Land disputes became clearly political and legal struggles with an ecological reference.Land disputes were stimulated when the common land/resource rights on village lands of the Maasai pastoralists became regulated and insecure. The analysis of past land disputes showed that space-place tensions on village lands can be presented as a platform on which spatial and property issues with complex power relations have been debated. The reduction of future land disputes will succeed only when/if local property rights to land and resources are acknowledged, especially in rural lands of the Tanzanian State.

Environmental effects of thermal and radioactive discharges from nuclear power plants in the boreal brackish-water conditions of the northern Baltic Sea

During recent decades, thermal and radioactive discharges from nuclear power plants into the aquatic environment have become the subject of lively debate as an ecological concern. The target of this thesis was to summarize the large quantity of results obtained in extensive monitoring programmes and studies carried out in recipient sea areas off the Finnish nuclear power plants at Loviisa and Olkiluoto during more than four decades. The Loviisa NPP is located on the coast of the Gulf of Finland and Olkiluoto NPP on that of the Bothnian Sea. The state of the Gulf of Finland is clearly more eutrophic; the nutrient concentrations in the surface water are about 1½ 2 times higher at Loviisa than at Olkiluoto, and the total phosphorus concentrations still increased in both areas (even doubled at Loviisa) between the early 1970s and 2000. Thus, it is a challenge to distinguish the local effects of thermal discharges from the general eutrophication process of the Gulf of Finland. The salinity is generally low in the brackish-water conditions of the northern Baltic Sea, being however about 1 higher at Olkiluoto than at Loviisa (the salinity of surface water varying at the latter from near to 0 in early spring to 4 6 in late autumn). Thus, many marine and fresh-water organisms live in the Loviisa area close to their limit of existence, which makes the biota sensitive to any additional stress. The characteristics of the discharge areas of the two sites differ from each other in many respects: the discharge area at Loviisa is a semi-enclosed bay in the inner archipelago, where the exchange of water is limited, while the discharge area at Olkiluoto is more open, and the exchange of water with the open Bothnian Sea is more effective. The effects of the cooling water discharged from the power plants on the temperatures in the sea were most obvious in winter. The formation of a permanent ice cover in the discharge areas has been delayed in early winter, and the break-up of the ice occurs earlier in spring. The prolonging of the growing season and the disturbance of the overwintering time, in conditions where the biota has adjusted to a distinct rest period in winter, have been the most significant biological effects of the thermal pollution. The soft-bottom macrofauna at Loviisa has deteriorated to the point of almost total extinction at many sampling stations during the past 40 years. A similar decline has been reported for the whole eastern Gulf of Finland. However, the local eutrophication process seems to have contributed into the decline of the zoobenthos in the discharge area at Loviisa. Thermal discharges have increased the production of organic matter, which again has led to more organic bottom deposits. These have in turn increased the tendency of the isolated deeps to a depletion of oxygen, and this has further caused strong remobilization of phosphorus from the bottom sediments. Phytoplankton primary production and primary production capacity doubled in the whole area between the late 1960s and the late 1990s, but started to decrease a little at the beginning of this century. The focus of the production shifted from spring to mid- and late summer. The general rise in the level of primary production was mainly due to the increase in nutrient concentrations over the whole Gulf of Finland, but the thermal discharge contributed to a stronger increase of production in the discharge area compared to that in the intake area. The eutrophication of littoral vegetation in the discharge area has been the most obvious, unambiguous and significant biological effect of the heated water. Myriophyllum spicatum, Potamogeton perfoliatus and Potamogeton pectinatus, and vigorous growths of numerous filamentous algae as their epiphytes have strongly increased in the vicinity of the cooling water outlet, where they have formed dense populations in the littoral zone in late summer. However, the strongest increase of phytobenthos has extended only to a distance of about 1 km from the outlet, i.e., the changes in vegetation have been largest in those areas that remain ice-free in winter. Similar trends were also discernible at Olkiluoto, but to a clearly smaller extent, which was due to the definitely weaker level of background eutrophy and nutrient concentrations in the Bothnian Sea, and the differing local hydrographical and biological factors prevailing in the Olkiluoto area. The level of primary production has also increased at Olkiluoto, but has remained at a clearly lower level than at Loviisa. In spite of the analogous changes observed in the macrozoobenthos, the benthic fauna has remained strong and diversified in the Olkiluoto area. Small amounts of local discharge nuclides were regularly detected in environmental samples taken from the discharge areas: tritium in seawater samples, and activation products, such as 60Co, 58Co, 54Mn, 110mAg, 51Cr, in suspended particulate matter, bottom sediments and in several indicator organisms (e.g., periphyton and Fucus vesiculosus) that effectively accumulate radioactive substances from the medium. The tritium discharges and the consequent detection frequency and concentrations of tritium in seawater were higher at Loviisa, but the concentrations of the activation products were higher at Olkiluoto, where traces of local discharge nuclides were also observed over a clearly wider area, due to the better exchange of water than at Loviisa, where local discharge nuclides were only detected outside Hästholmsfjärden Bay quite rarely and in smaller amounts. At the farthest, an insignificant trace amount (0.2 Bq kg-1 d.w.) of 60Co originating from Olkiluoto was detected in Fucus at a distance of 137 km from the power plant. Discharge nuclides from the local nuclear power plants were almost exclusively detected at the lower trophic levels of the ecosystems. Traces of local discharge nuclides were very seldom detected in fish, and even then only in very low quantities. As a consequence of the reduced discharges, the concentrations of local discharge nuclides in the environment have decreased noticeably in recent years at both Loviisa and Olkiluoto. Although the concentrations in environmental samples, and above all, the discharge data, are presented as seemingly large numbers, the radiation doses caused by them to the population and to the biota are very low, practically insignificant. The effects of the thermal discharges have been more significant, at least to the wildlife in the discharge areas of the cooling water, although the area of impact has been relatively small. The results show that the nutrient level and the exchange of water in the discharge area of a nuclear power plant are of crucial importance.

A study on the variations of optical and physical properties of aerosols over a tropical semi-arid station during grassland fire

The present paper records the results of a case study on the impact of an extensive grassland fire on the physical and optical properties of aerosols at a semi-arid station in southern India for the first time from ground based measurements using a MICROTOPS-II sunphotometer, an aethalometer and a quartz crystal microbalance impactor (QCM). Observations revealed a substantial increase in aerosol optical depth (AOD) at all wavelengths during burning days compared to normal days. High AOD values observed at shorter wavelengths suggest the dominance of accumulation mode particle loading over the study area. Daily mean aerosol size spectra shows, most of the time, power-law distribution. To characterize AOD, the Angstrom parameters (i.e., alpha and beta) were used. Wavelength exponent (1.38) and turbidity coefficient (0.21) are high during burning days compared to normal days, thereby suggesting an increase in accumulation mode particle loading. Aerosol size distribution suggested dominance of accumulation mode particle loading during burning days compared to normal days. A significant positive correlation was observed between AOD at 500 mn and water vapour and negative correlation between AOD at 500 nm and wind speed for burning and non-burning days. Diurnal variations of black carbon (BC) aerosol mass concentrations increased by a factor of similar to 2 in the morning and afternoon hours during burning period compared to normal days.

Sensor network deployment for agronomical data gathering in semi-arid regions

We share our experience in planning, designing and deploying a wireless sensor network of one square kilometre area. Environmental data such as soil moisture, temperature, barometric pressure, and relative humidity are collected in this area situated in the semi-arid region of Karnataka, India. It is a hope that information derived from this data will benefit the marginal farmer towards improving his farming practices. Soon after establishing the need for such a project, we begin by showing the big picture of such a data gathering network, the software architecture we have used, the range measurements needed for determining the sensor density, and the packaging issues that seem to play a crucial role in field deployments. Our field deployment experiences include designing with intermittent grid power, enhancing software tools to aid quicker and effective deployment, and flash memory corruption. The first results on data gathering look encouraging.

Localized data gathering paradigms for small and marginal farm lands in Semi-Arid region – Issues and Concerns

In this work we explore the application of wireless sensor technologies for the benefit of small and marginal farmers in semi-arid regions. The focus in this paper is to discuss the merits and demerits of data gathering & relay paradigms that collect localized data over a wide area. The data gathered includes soil moisture, temperature, pressure, rain data and humidity. The challenge to technology intervention comes mainly due to two reasons: (a) Farmers in general are interested in crop yield specific to their piece of land. This is because soil texture can vary rapidly over small regions. (b) Due to a high run-off, the soil moisture retention can vary from region to region depending on the topology of the farm. Both these reasons alter the needs drastically. Additionally, small and marginal farms can be sandwiched between rich farm lands. The village has very little access to grid power. Power cuts can extend up to 12 hours in a day and upto 3 or 4 days during some months in the year. In this paper, we discuss 3 technology paradigms for data relaying. These include Wi-Fi (Wireless Fidelity), GPRS (General Packet Radio Service) and DTN (Delay and Disruption Tolerant Network) technologies. We detail the merits and demerits of each of these solutions and provide our final recommendations. The project site is a village called Chennakesavapura in the state of Karnataka, India.

Minimum power consumption for drag reduction on a circular cylinder by tangential surface motion

We investigate the effect of a prescribed tangential velocity on the drag force on a circular cylinder in a spanwise uniform cross flow. Using a combination of theoretical and numerical techniques we make an attempt at determining the optimal tangential velocity profiles which will reduce the drag force acting on the cylindrical body while minimizing the net power consumption characterized through a non-dimensional power loss coefficient (C-PL). A striking conclusion of our analysis is that the tangential velocity associated with the potential flow, which completely suppresses the drag force, is not optimal for both small and large, but finite Reynolds number. When inertial effects are negligible (R e << 1), theoretical analysis based on two-dimensional Oseen equations gives us the optimal tangential velocity profile which leads to energetically efficient drag reduction. Furthermore, in the limit of zero Reynolds number (Re -> 0), minimum power loss is achieved for a tangential velocity profile corresponding to a shear-free perfect slip boundary. At finite Re, results from numerical simulations indicate that perfect slip is not optimum and a further reduction in drag can be achieved for reduced power consumption. A gradual increase in the strength of a tangential velocity which involves only the first reflectionally symmetric mode leads to a monotonic reduction in drag and eventual thrust production. Simulations reveal the existence of an optimal strength for which the power consumption attains a minima. At a Reynolds number of 100, minimum value of the power loss coefficient (C-PL = 0.37) is obtained when the maximum in tangential surface velocity is about one and a half times the free stream uniform velocity corresponding to a percentage drag reduction of approximately 77 %; C-PL = 0.42 and 0.50 for perfect slip and potential flow cases, respectively. Our results suggest that potential flow tangential velocity enables energetically efficient propulsion at all Reynolds numbers but optimal drag reduction only for Re -> infinity. The two-dimensional strategy of reducing drag while minimizing net power consumption is shown to be effective in three dimensions via numerical simulation of flow past an infinite circular cylinder at a Reynolds number of 300. Finally a strategy of reducing drag, suitable for practical implementation and amenable to experimental testing, through piecewise constant tangential velocities distributed along the cylinder periphery is proposed and analysed.

Medium-Voltage Drive for Induction Machine With Multilevel Dodecagonal Voltage Space Vectors With Symmetric Triangles

Multilevel inverters with dodecagonal (12-sided polygon) voltage space vector structure have advantages, such as complete elimination of fifth and seventh harmonics, reduction in electromagnetic interference, reduction in device voltage ratings, reduction of switching frequency, extension of linear modulation range, etc., making it a viable option for high-power medium-voltage drives. This paper proposes two power circuit topologies capable of generating multilevel dodecagonal voltage space vector structure with symmetric triangles (for the first time) with minimum number of dc-link power supplies and floating capacitor H-bridges. The first power topology is composed of two hybrid cascaded five-level inverters connected to either side of an open-end winding induction machine. Each inverter consists of a three-level neutral-point-clamped inverter, which is cascaded with an isolated H-bridge making it a five-level inverter. The second topology is for a normal induction motor. Both of these circuit topologies have inherent capacitor balancing for floating H-bridges for all modulation indexes, including transient operations. The proposed topologies do not require any precharging circuitry for startup. A simple pulsewidth modulation timing calculation method for space vector modulation is also presented in this paper. Due to the symmetric arrangement of congruent triangles within the voltage space vector structure, the timing computation requires only the sampled reference values and does not require any offline computation, lookup tables, or angle computation. Experimental results for steady-state operation and transient operation are also presented to validate the proposed concept.

An Open-End winding IM drive with Multilevel 12-sided Polygonal Vectors with Symmetric Triangles

Multilevel inverters with hexagonal voltage space vector structures have improved performance of induction motor drives compared to that of the two level inverters. Further reduction in the torque ripple on the motor shaft is possible by using multilevel dodecagonal (12-sided polygon) voltage space vector structures. The advantages of dodecagonal voltage space vector based PWM techniques are the complete elimination of fifth and seventh harmonics in phase voltages for the full modulation range and the extension of linear modulation range. This paper proposes an inverter circuit topology capable of generating multilevel dodecagonal voltage space vectors with symmetric triangles, by cascading two asymmetric three level inverters with isolated H-Bridges. This is made possible by proper selection of DC link voltages and the selection of resultant switching states for the inverters. In this paper, a simple PWM timing calculation method is proposed. Experimental results have also been presented in this paper to validate the proposed concept.

Semi-gas kinetics model for performance modeling of flowing chemical oxygen-iodine lasers (COIL)

A semi-gas kinetics (SGK) model for performance analyses of flowing chemical oxygen-iodine laser (COIL) is presented. In this model, the oxygen-iodine reaction gas flow is treated as a continuous medium, and the effect of thermal motions of particles of different laser energy levels on the performances of the COIL is included and the velocity distribution function equations are solved by using the double-parameter perturbational method. For a premixed flow, effects of different chemical reaction systems, different gain saturation models and temperature, pressure, yield of excited oxygen, iodine concentration and frequency-shift on the performances of the COIL are computed, and the calculated output power agrees well with the experimental data. The results indicate that the power extraction of the SGK model considering 21 reactions is close to those when only the reversible pumping reaction is considered, while different gain saturation models and adjustable parameters greatly affect the output power, the optimal threshold gain range, and the length of power extraction.

Singular behaviour near the tip of a sharp V-notch in a power law hardening material

This paper presents an asymptotic analysis of the near-tip stress and strain fields of a sharp V-notch in a power law hardening material. First, the asymptotic solutions of the HRR type are obtained for the plane stress problem under symmetric loading. It is found that the angular distribution function of the radial stress sigma(r) presents rapid variation with the polar angle if the notch angle beta is smaller than a critical notch angle; otherwise, there is no such phenomena. Secondly, the asymptotic solutions are developed for antisymmetric loading in the cases of plane strain and plane stress. The accurate calculation results and the detailed comparisons are given as well. All results show that the singular exponent s is changeable for various combinations of loading condition and plane problem.

Em busca do convívio social: o regime semi-aberto no Instituto Penal Oscar Stevenson

O presente trabalho tem como objetivo central analisar o Sistema Penitenciário do Estado do Rio de Janeiro a partir do regime semi-aberto, tendo como campo de análise o Instituto Penal Oscar Stevenson, situado em Benfica, no município do Rio de Janeiro, voltado para um público carcerário feminino. Buscou-se verificar, sob o enfoque das presas, a expectativa e possibilidades de retorno ao convívio social; analisar os aspectos jurídico-institucionais referentes ao regime semi-aberto, no que tange a obtenção dos benefícios, junto a Lei de Execução Penal e identificar quais as parcerias que viabilizam a inserção delas no mercado de trabalho. Para a efetivação desse trabalho utilizou-se, preferencialmente os pressupostos teóricos e metodológicos da pesquisa quali-quantitativa, pois foi trabalhado não só no nível da objetividade, mas também no significado das ações e relações humanas, sabendo que a realidade prisional é perpassada por questões de cunho opressor, punitivo, em função de preconizar a segurança. Foram realizados também levantamentos de dados bibliográficos e censitários, bem como entrevistas semi-estruturadas junto aos agentes penitenciários do setor de educação e classificação e principalmente as presas. A análise do material coletado permitiu confirmar as hipóteses da pesquisa: i) que a ausência de oportunidades que garantam às presas os benefícios do regime semi-aberto não se dá por falta de instrumentos legais, mas sim pela burocracia no cadastramento e poucas parcerias de cursos profissionalizantes, empresas privadas que absorvam mão-de-obra das presas do regime semi-aberto; e ii) e que no momento em que as presas ainda estavam no regime fechado, não tiveram oportunidades de se capacitarem e também os vínculos familiares não foram mantidos, com isso dificultando que estas usufruam dos benefícios do regime semi-aberto. E, conseqüentemente, sendo cada vez mais adiado o seu retorno gradativo ao convívio social, através da progressão de regime.

Distributed Control and Optimization for Communication and Power Systems

We are at the cusp of a historic transformation of both communication system and electricity system. This creates challenges as well as opportunities for the study of networked systems. Problems of these systems typically involve a huge number of end points that require intelligent coordination in a distributed manner. In this thesis, we develop models, theories, and scalable distributed optimization and control algorithms to overcome these challenges.

This thesis focuses on two specific areas: multi-path TCP (Transmission Control Protocol) and electricity distribution system operation and control. Multi-path TCP (MP-TCP) is a TCP extension that allows a single data stream to be split across multiple paths. MP-TCP has the potential to greatly improve reliability as well as efficiency of communication devices. We propose a fluid model for a large class of MP-TCP algorithms and identify design criteria that guarantee the existence, uniqueness, and stability of system equilibrium. We clarify how algorithm parameters impact TCP-friendliness, responsiveness, and window oscillation and demonstrate an inevitable tradeoff among these properties. We discuss the implications of these properties on the behavior of existing algorithms and motivate a new algorithm Balia (balanced linked adaptation) which generalizes existing algorithms and strikes a good balance among TCP-friendliness, responsiveness, and window oscillation. We have implemented Balia in the Linux kernel. We use our prototype to compare the new proposed algorithm Balia with existing MP-TCP algorithms.

Our second focus is on designing computationally efficient algorithms for electricity distribution system operation and control. First, we develop efficient algorithms for feeder reconfiguration in distribution networks. The feeder reconfiguration problem chooses the on/off status of the switches in a distribution network in order to minimize a certain cost such as power loss. It is a mixed integer nonlinear program and hence hard to solve. We propose a heuristic algorithm that is based on the recently developed convex relaxation of the optimal power flow problem. The algorithm is efficient and can successfully computes an optimal configuration on all networks that we have tested. Moreover we prove that the algorithm solves the feeder reconfiguration problem optimally under certain conditions. We also propose a more efficient algorithm and it incurs a loss in optimality of less than 3% on the test networks.

Second, we develop efficient distributed algorithms that solve the optimal power flow (OPF) problem on distribution networks. The OPF problem determines a network operating point that minimizes a certain objective such as generation cost or power loss. Traditionally OPF is solved in a centralized manner. With increasing penetration of volatile renewable energy resources in distribution systems, we need faster and distributed solutions for real-time feedback control. This is difficult because power flow equations are nonlinear and kirchhoff's law is global. We propose solutions for both balanced and unbalanced radial distribution networks. They exploit recent results that suggest solving for a globally optimal solution of OPF over a radial network through a second-order cone program (SOCP) or semi-definite program (SDP) relaxation. Our distributed algorithms are based on the alternating direction method of multiplier (ADMM), but unlike standard ADMM-based distributed OPF algorithms that require solving optimization subproblems using iterative methods, the proposed solutions exploit the problem structure that greatly reduce the computation time. Specifically, for balanced networks, our decomposition allows us to derive closed form solutions for these subproblems and it speeds up the convergence by 1000x times in simulations. For unbalanced networks, the subproblems reduce to either closed form solutions or eigenvalue problems whose size remains constant as the network scales up and computation time is reduced by 100x compared with iterative methods.

Elementos finitos em fluidos dominados pelo fenômeno de advecção: um método semi-Lagrangeano.

Os escoamentos altamente convectivos representam um desafio na simulação pelo método de elementos finitos. Com a solução de elementos finitos de Galerkin para escoamentos incompressíveis, a matriz associada ao termo convectivo é não simétrica, e portanto, a propiedade de aproximação ótima é perdida. Na prática as soluções apresentam oscilações espúrias. Muitos métodos foram desenvolvidos com o fim de resolver esse problema. Neste trabalho apresentamos um método semi- Lagrangeano, o qual é implicitamente um método do tipo upwind, que portanto resolve o problema anterior, e comparamos o desempenho do método na solução das equações de convecção-difusão e Navier-Stokes incompressível com o Streamline Upwind Petrov Galerkin (SUPG), um método estabilizador de reconhecido desempenho. No SUPG, as funções de forma e de teste são tomadas em espaços diferentes, criando um efeito tal que as oscilações espúrias são drasticamente atenuadas. O método semi-Lagrangeano é um método de fator de integração, no qual o fator é um operador de convecção que se desloca para um sistema de coordenadas móveis no fluido, mas restabelece o sistema de coordenadas Lagrangeanas depois de cada passo de tempo. Isto prevê estabilidade e a possibilidade de utilizar passos de tempo maiores.Existem muitos trabalhos na literatura analisando métodos estabilizadores, mas não assim com o método semi-Lagrangeano, o que representa a contribuição principal deste trabalho: reconhecer as virtudes e as fraquezas do método semi-Lagrangeano em escoamentos dominados pelo fenômeno de convecção.

Multimode radiation from an unflanged, semi-infinite circular duct with uniform flow.

Multimode sound radiation from an unflanged, semi-infinite, rigid-walled circular duct with uniform subsonic mean flow everywhere is investigated theoretically. The multimode directivity depends on the amplitude and directivity function of each individual cut-on mode. The amplitude of each mode is expressed as a function of cut-on ratio for a uniform distribution of incoherent monopoles, a uniform distribution of incoherent axial dipoles, and for equal power per mode. The directivity function of each mode is obtained by applying a Lorentz transformation to the zero-flow directivity function, which is given by a Wiener-Hopf solution. This exact numerical result is compared to an analytic solution, valid in the high-frequency limit, for multimode directivity with uniform flow. The high-frequency asymptotic solution is derived assuming total transmission of power at the open end of the duct, and gives the multimode directivity function with flow in the forward arc for a general family of mode amplitude distribution functions. At high frequencies the agreement between the exact and asymptotic solutions is shown to be excellent.