Geometry in Nature by Vagn Lundsgaard Hansen

By Vagn Lundsgaard Hansen

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Small vibrations. 2. A derivation of t h e wave equation The wave equation will now be derived by applying Newton's Second Law of Motion to a piece of the string. 2). 2. A derivation of t h e wave equation . 2. A piece S of the string. 1) (Mass of S) (Acceleration of S) = Net force acting on S where acceleration and force are in a direction perpendicular to the x—axis. The next step is to calculate the mass, acceleration, and net force acting on S. The mass of S is the string density p times the length of 5, so rX-\-Ax Mass of S = p • / y/l + Jx {ux(s,t))2ds.

Waves represented by functions of the form u(x,t) — f(x — ct) are called traveling waves. The two basic features of any traveling wave are the underlying profile shape defined by / and the speed \c\ 23 4. Traveling Waves 24 at which the profile is translated along the x—axis. It is assumed that the function / is not constant and c is not zero in order for u(x, t) to represent the movement of a disturbance through a medium. 1. The function u(x,t) e (x 5t) r e p r e s e n t s a travelmoving in the positive x ing wave with initial profile u(x,0) = e direction with speed 5.

Acting on the left end of S. The vertical component of this force is then T y/l + ux(x,t) (ux(x,t))*' Under the assumption of small vibrations, we again make the approximation y/l -f (ux)2 « 1, and so the vertical component of the force due to tension on the left side of S is approximately -Tux(x,t). Repeating this construction at the right end x + Ax of 5, the vertical component of the force due to tension on the right side of S is approximately Tux(x + Ax,t). 3) Net force on S = Tux{x + Ax, i) - Tux(x, t).

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