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- W2739415775 abstract "In 1968 Roddam Narasimha (RN) published a paper in Journal of Sound and Vibration (JSV) deriving the equation $$frac{{{partial ^2}v}} {{partial {t^2}}} + 2Rfrac{{partial v}} {{partial t}} = left[ {1 + frac{1} {2}Gamma 'intlimits_0^l {v_x^2dx} } right]frac{{{partial ^2}v}} {{partial {t^2}}} + {f_0}left( {x,t} right)$$ ((1.1)) for the transverse displacement v(x, t) of a vibrating string of length l, where v = (v, w) is a two-dimensional vector in the yz plane orthogonal to the x axis, (v_x^2 = v_x^2 + w_x^2) is the squared x derivative of v, R is a damping coefficient, Γ′ a nonlinearity parameter that involves a characteristic amplitude of the string motion and the material properties of the string (see equation 4.25 below) and f0 is an external force acting on the string. Note that for R ≡ 0 and Γ′ ≡ 0 this is the standard linear wave equation that is the canonical example for a second order hyperbolic equation in one space dimension which ignores the coupling (usually nonlinear) between the transverse and longitudinal displacements of the string. In the usual text book derivation of the wave equation for the vibrating string it is assumed that the motion of the string is entirely transversal. That this is not strictly true was realized by Kirchhoff [1876] and Lord Rayleigh in the nineteenth century: while the former went on to derive an equation similar to (1.1), Rayleigh [1883] (with no reference to Kirchhoff) restricted himself to the classical van der Pol oscillator to model the vibrating string. More such models were proposed by Osgood in the 1920’s, Carrier and Coulson in the 1940’s, followed by Oplinger, Murthy and Ramakrishna, Narasimha and Anand in the 1960’s." @default.
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- W2739415775 date "2013-01-01" @default.
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- W2739415775 title "On the String Equation of Narasimha" @default.
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