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

4.10 Forced Vibrations in an Infinite String 81nth mode of vibration isE n = m 4 ω2 n A2 n = m 4( nπcL) ( ) 2 8d 2= 16md2 c 2n 2 π 2 n 2 π 2 L 2It becomes apparent that as n increases, the energy of the nth mode lessens. Forexample, the energy of the third harmonic is one-ninth that of the fundamentalmode.4.10 Forced Vibrations in an Infinite StringWhile this may appear to be a purely academic exercise, the simple case of atransverse sinusoidal force on an idealized string of infinite length can provideinsight into the forced vibrations of finite strings and the transmission of acousticwaves.An ideal string of infinite length subject to a tension T receives a transversedriving force T cos ωt at the string end x = 0. The end at x = 4 is rigidly clampedbut the point x = 0 is rigid only in the x-direction, being free to move in they-direction, a support that can be approximated by a pivoted lever as shown inFigure 4.5. Thus, the driving force can move the lever as well as the string. We shallneglect the mechanical impedance of the pivoted lever (or hinge) that is deemedto have no friction and no stiffness. No waves are reflected from the far end x = 4,and hence no waves travel in the negative x-direction. The displacement of thestring can now be described by the general solution containing only the expressionfor a harmonic wave traveling in the positive x-direction:or in a complex format:y = a 1 sin(ωt − kx) + b1 cos(ωt − kx)y = A e i(ωt−kx) (4.21)Figure 4.5. Forces acting at one end of an infinite string.