title: Lie theory and the wave equation in space–time. 5. R-separable solutions of the wave equation ψtt-∆3ψ=0

authors: Kalnins, Ernie G.

subjects: Mathematics

abstract: A detailed classification is made of all orthogonal coordinate systems for which the wave equation ψtt-∆3ψ=0 admits an R-separable solution. Only those coordinate systems are given which are not conformally equivalent to coordinate systems that have already been found in previous articles. We find 106 coordinates to given a total of 368 conformally inequivalent orthogonal coordinates for which the wave equation admits an R separation of variables.

text: Lie theory and the wave equation in space–time. 5. R-separable solutions of the wave equation ψtt-∆3ψ=0 A detailed classification is made of all orthogonal coordinate systems for which the wave equation ψtt-∆3ψ=0 admits an R-separable solution. Only those coordinate systems are given which are not conformally equivalent to coordinate systems that have already been found in previous articles. We find 106 coordinates to given a total of 368 conformally inequivalent orthogonal coordinates for which the wave equation admits an R separation of variables.

year: 1978-06

title: Wave-frequency flows within a near-bed vegetation canopy

authors: Henderson, Stephen M.

subjects: Science & Technology

abstract: We study water flows and wave dissipation within near-bed pneumatophore canopies at the wave-exposed fringe of a mangrove forest on Cù Lao Dung Island, in the Mekong Delta. To evaluate canopy drag, the three-dimensional geometry of pneumatophore stems growing upward from the buried lateral roots of Sonneratia caseolaris mangroves was reconstructed from photogrammetric surveys. In cases where hydrodynamic measurements were obtained, up to 84 stems per square meter were observed, with stem heights <  0.6 m, and basal diameters 0.01–0.02 m. The parameter α =(frontal area of pneumatophores blocking the flow)/(canopy volume) ranged from zero to 1.8m−¹. Within-canopy water velocity displayed a phase lead and slight attenuation relative to above-canopy flows. The phase lead was frequency-dependent, ranging from 0 to 30 degrees at the frequencies of energetic waves ( >0.1 Hz), and up to 90 degrees at lower frequencies. A model is developed for wave-induced flows within the vertically variable canopy. Scaling suggests that acceleration-induced forces and vertical mixing were negligible at wave frequencies. Consistent with theory, drag-induced vertical variability in velocity scaled with ꓥ = Tw/(2πTf) , where Tw=wave period, Tf=2/(CD₍ₐ|ᵤ|₎) is the frictional time scale, CD≈2 is the drag coefficient, and  |ᵤ| is a typical flow speed. For fixed wave conditions ( |ᵤ| and Tw), theory predicts increasing dissipation with increasing vegetation density (i.e. increasing a), until a maximum is reached for order-one Λ. For larger Λ, within-canopy flow is so inhibited by drag that further increases in a reduce within-canopy dissipation. For observed cases, Λ≤0.38 at energetic wave frequencies, so wave dissipation near the forest edge is expected to increase with increasing pneumatophore canopy density. However, under different wave conditions, the most dense canopies may occasionally approach the dissipation maximum (Λ≈1). Predicted dissipation by the pneumatophore canopy was sufficient to attenuate most wave energy over distances slightly less (more) than 100 m into the marsh in 1 m (2 m) water depth.

text: Wave-frequency flows within a near-bed vegetation canopy We study water flows and wave dissipation within near-bed pneumatophore canopies at the wave-exposed fringe of a mangrove forest on Cù Lao Dung Island, in the Mekong Delta. To evaluate canopy drag, the three-dimensional geometry of pneumatophore stems growing upward from the buried lateral roots of Sonneratia caseolaris mangroves was reconstructed from photogrammetric surveys. In cases where hydrodynamic measurements were obtained, up to 84 stems per square meter were observed, with stem heights <  0.6 m, and basal diameters 0.01–0.02 m. The parameter α =(frontal area of pneumatophores blocking the flow)/(canopy volume) ranged from zero to 1.8m−¹. Within-canopy water velocity displayed a phase lead and slight attenuation relative to above-canopy flows. The phase lead was frequency-dependent, ranging from 0 to 30 degrees at the frequencies of energetic waves ( >0.1 Hz), and up to 90 degrees at lower frequencies. A model is developed for wave-induced flows within the vertically variable canopy. Scaling suggests that acceleration-induced forces and vertical mixing were negligible at wave frequencies. Consistent with theory, drag-induced vertical variability in velocity scaled with ꓥ = Tw/(2πTf) , where Tw=wave period, Tf=2/(CD₍ₐ|ᵤ|₎) is the frictional time scale, CD≈2 is the drag coefficient, and  |ᵤ| is a typical flow speed. For fixed wave conditions ( |ᵤ| and Tw), theory predicts increasing dissipation with increasing vegetation density (i.e. increasing a), until a maximum is reached for order-one Λ. For larger Λ, within-canopy flow is so inhibited by drag that further increases in a reduce within-canopy dissipation. For observed cases, Λ≤0.38 at energetic wave frequencies, so wave dissipation near the forest edge is expected to increase with increasing pneumatophore canopy density. However, under different wave conditions, the most dense canopies may occasionally approach the dissipation maximum (Λ≈1). Predicted dissipation by the pneumatophore canopy was sufficient to attenuate most wave energy over distances slightly less (more) than 100 m into the marsh in 1 m (2 m) water depth.

year: 2017

title: Contextualising social enterprise in New Zealand

authors: Grant, Suzanne Lisa Parker

subjects: business enterprise

abstract: Purpose – Social enterprise in New Zealand is still in its infancy, with no recognised framework to inform knowledge of current or future developments. In this exploratory paper, the aim is to consider four influences which are shaping the development of social enterprise in New Zealand.

Design/methodology/approach – A critical-appreciative lens utilising Habermas' concepts of the lifeworld and system informs the consideration of these influences.

Findings – Four distinct cultural and historical influences are proposed as contributing to the scope and “flavour” of social enterprise developing in New Zealand: socio-cultural norms, e.g. “Kiwi ingenuity”; the neoliberal reforms initiated by successive governments during the 1980s; Crown settlements in relation to breaches of the principles of 1840 Treaty of Waitangi; New Zealanders' as international citizens.

Originality/value – The paper shows how feedback and dialogue across the sectors, at local, national and international levels, is now required to determine how other scholars, practitioners and policy makers perceive this proposed initial framework.

text: Contextualising social enterprise in New Zealand Purpose – Social enterprise in New Zealand is still in its infancy, with no recognised framework to inform knowledge of current or future developments. In this exploratory paper, the aim is to consider four influences which are shaping the development of social enterprise in New Zealand.

Design/methodology/approach – A critical-appreciative lens utilising Habermas' concepts of the lifeworld and system informs the consideration of these influences.

Findings – Four distinct cultural and historical influences are proposed as contributing to the scope and “flavour” of social enterprise developing in New Zealand: socio-cultural norms, e.g. “Kiwi ingenuity”; the neoliberal reforms initiated by successive governments during the 1980s; Crown settlements in relation to breaches of the principles of 1840 Treaty of Waitangi; New Zealanders' as international citizens.

Originality/value – The paper shows how feedback and dialogue across the sectors, at local, national and international levels, is now required to determine how other scholars, practitioners and policy makers perceive this proposed initial framework.

year: 2008