THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

78 4. Vibrating StringsFigure 4.4. The sixth harmonic mode of a vibrating string stretched between x = 0 andx = L.where f n is the frequency of vibration in the nth mode and the wavelength constantby k n = 2π/λ n . From Equation (4.10) we deriveλ n = 2L/n (4.13)that is, the wavelength is twice the nodal distance of the associated wave pattern.4.8 The Effect of Initial ConditionsThe complete general solution to the general harmonic wave equation for a freelyvibrating string rigidly clamped at its ends contains all the individual modes ofvibration described by Equation (4.12). It is expressed as∞∑y n = (An cos ω n t + Bn sin ω n t) sin k n t (4.14)n=1where A n and B n are the amplitude coefficients dependent on the method of excitingthe string to vibrate. The actual amplitude of the nth mode is√a n = A 2 n + B2 nConsider the initial condition at t = 0 at the time when the string is displaced fromits normal linear configuration so that the displacement y(x,t) at each point of thestring is given by the functiony(x, 0) = y 0 (x)The corresponding velocity v(x,t) is given for t = 0byv(x, 0) =∂y(x, 0)∂t= v 0 (x)