Under consideration for publication in J. Fluid Mech. 1 Absolute/Convective instability of a flapping flag By SANKHA BANERJEE, AND DICK K. P. YUE† Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Received 3 July 2013) The work presents the study of the nature of instability, whether absolute or convec- tive of a two-dimensional flapping filament submerged in a uniform inflow. When the structure-to-fluid mass ratio, μ = ρ s h/(ρ f L) is zero, two waves equal in magnitude and opposite signs exist. We show by constructing an energy conservation law for the lin- earized equations of motion that both types of waves have initially positive energy in the sense that they both require an energy source to be sustained, after a critical value of μ is exceeded the slow waves acquire negative energy, requiring an energy sink for sustainance. With increasing μ for fixed Re = V L/ν = 1000, a flapping instability is sustained by the coalescence between the fast (positive energy) and the slow (negative energy) waves creating waves with zero energy which do not require neither an energy source nor a sink to be sustained, and grow exponentially in time. For almost all values of the μ and Re , the instability was found absolute except a narrow range of μ 0.060  μ  0.075 over which convective instability prevailed. This unstable flapping amplitude at the instability threshold is found to satisfy the linearly unstable Klein-Gordon equation. 1. Introduction As emphasized in a recent review Shelley & Zhang (2011), the flapping of flag is consid- ered as a canonical problem for studying the instability propagation in numerous physical systems where the dynamics of a slender structure is coupled to an axial flow. Landahl (1962) and Benjamin (1963), started such analysis giving rise to a profusion of instability types in simple physical models. This theoretical framework, was introduced in hydrody- namics by Huerre & Monkewitz (1990), and has been fruitfully applied to fluid-structure interactions problems. The generic case of inviscid elastic plate, further referred to as the flat-plate problem, has been extensively analyzed (Brazier-Smith & Scott (1984);Carpen- ter & Garrad (1986);Crighton et al. (1991); Lucey & Carpenter (1992); Peake (2003)) and extended to membranes under tension by ?). These analyses have brought to light many of the fundamental features of these interactions, such as the existence of negative energy waves or possible violations of the usual out-going wave radiation condition. Two different approaches are commonly used to study the instability properties of such systems. The first approach considers the medium to be of infinite length, in this case the waves propagating in the medium are considered through the analysis of the local wave equation. If temporally amplified waves are identified the medium is said to be locally unstable. Depending on the long time impulse response of the locally unstable medium, two types of instabilities may be distinguished: convective or absolute. The second ap- proach considers the same medium, but of finite length. The modes are studied, through † Author to whom correspondence should be addressed: yue@mit.edu