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9.2: The Wave Equation

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  • As with all phenomena in classical mechanics, the motion of the particles in a wave, for instance the masses on springs in figure 9.1, are governed by Newton’s laws of motion and the various force laws. In this section we will use these laws to derive an equation of motion for the wave itself, which applies quite generally to wave phenomena. To do so, consider a series of particles of equal mass \(m\) connected by springs of spring constant \(k\), again as in figure 9.1a, and assume that at rest the distance between any two masses ish. Let theposition of particleibex, and \(u\) the distance that particle is away from its rest position; thenu∆xrest°xisa function of both position \(x\) and time \(t\). Suppose particleihas moved to the left, then it will feel a restoring force to the right due to two sources: the compressed spring on its left, and the extended spring on its right.The total force to the right is then given by:

    \[\begin{aligned} F_{i} &=F_{i+1 \rightarrow i}-F_{i-1 \rightarrow i} \\ &=k[u(x+h, t)-u(x, t)]-k[u(x, t)-u(x-h, t)] \\ &=k[u(x+h, t)-2 u(x, t)+u(x-h, t)] \end{aligned} \label{9.3}\]

    Equation \ref{9.3} gives the net force on particlei, which by Newton’s second law of motion (equation 2.5) equals the particle’s mass times its acceleration. The acceleration is the second time derivative of the positionx,but since the equilibrium position is a constant, it is also the second time derivative of the distance from theequilibrium positionu(x,t), and we have:

    \[F_{\mathrm{net}}=m \frac{\partial^{2} u(x, t)}{\partial t^{2}}=k[u(x+h, t)-2 u(x, t)+u(x-h, t)] \label{9.4}\]

    Equation \ref{9.4} holds for particle \(i\), but just as well for particle \(i+1\), or \(i-10\). We can get an equation for \(N\) particles by simply adding their individual equations, which we can do because these equations are linear.We thus find for a string of particles of length \(L=Nh\) hand total mass \(M=N m\):

    \[\frac{\partial^{2} u(x, t)}{\partial t^{2}}=\frac{K L^{2}}{M} \frac{u(x+h, t)-2 u(x, t)+u(x-h, t)}{h^{2}} \label{9.5}.\]