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

4.6 Simple Harmonic Solutions of the Wave Equation 75In this case, the wave traveling in the positive x-direction reflects back as a similarwave of opposite displacement moving in the negative x-direction. The majorresult of these two reflections is that the motion of the free vibration becomesperiodic. A pulse leaving x = 0 reaches x = L after an interval of L/c seconds.There, it is reflected and returns to the origin where it again undergoes a reflectionafter a time lapse of 2L/c seconds. The shape of the pulse after its second reflectionis identical with that of the original pulse. This periodicity has resulted from thespecified boundary conditions, i.e., fixed points at x = 0 and x = L.4.6 Simple Harmonic Solutions of the Wave EquationSimple harmonic vibrations frequently occur in nature, and we shall now considera simple harmonic motion (SHM) propagating along a string. Any vibration ofthe string, however complex, can be resolved into an equivalent array of simpleharmonic vibrations. This resolution of complex vibration into a series of SHMsis not a mere mathematical exercise but constitutes the phenomenal principle ofhow the ear functions. The ear breaks down a complex sound into its simpleharmonic components. This capability permits us to distinguish the differencesbetween different voices and musical instruments. A piano sounding a note willsound differently from the same note played by an oboe. If all the frequenciespresent in the sound consist of a fundamental tone plus its harmonics, they willsound more harmonious than in the situation where the frequencies are not relatedso simply to each other.The displacement of any point on the string exciting a SHM of angular frequencyω can be depicted by the special solution to Equation (4.3):y = a 1 sin(ωt − kx) + a 2 sin(ωt + kx) + b 1 cos(ωt − kx) + b 2 cos(ωt + kx)(4.7)where a 1 , a 2 , b 1 , b 2 are arbitrary constants and k is the wavelength constant givenbyk ≡ ω/cApplying the boundary condition y(0, t) = 0 (i.e., y = 0atx = 0), which describesa fixed point, Equation (4.7) reduces to(a 1 + a 2 ) sin ωt =−(b 1 + b 2 ) cos ωt (4.8)Because this equation applies to all values of t, the following relations betweenthe constants must exista 1 + a 2 = 0, b 1 + b 2 = 0 or a 1 =−a 2 , b 1 =−b 2The two limitations for the arbitrary constants of Equation (4.7) are equivalentto the single restriction of Equation (4.5) in the general solution of wave equation(4.3). The two waves must be of equal and opposite displacements and must