D'Alembert's Wave Equation

Figure: t=0, Figures 3 to 10 show an example of the traveling wave solution to a string that is displaced to an amplitude of $2$ at time t=0. We note that the string is held in place at three points, P1, P2 and P3 at t=0. The top plot of each figure shows the right traveling wave, the middle plot the left traveling wave and the bottom plot the physical displacement, equal to the sum of the top and middle plots. Plots generated using scripts developed by Ed Berdahl.

Figure 4: t=5

Figure 5: t=10

Figure 6: t=15

Figure 7: t=20

Figure 8: t=25

Figure 9: t=30

Figure 10: t=35

The formulation of the Wave Equation and its solution by d'Alembert in 1747 [1] is the theoretical starting-point for physical stringed models. The wave equation, written as

$$K\frac{\partial^2y}{\partial x^2} = \varepsilon\frac{\partial^2y}{\partial t^2}$$ (1)

where $K$ is the string tension, $\varepsilon$ is the linear mass density and $y(t,x)$ is the string displacement as a function of time ($t$ ) and position along the string ($x$ ). It can be derived directly from Newton's second law applied to a differential string element. In addition to introducing the 1D wave equation, d'Alembert introduced its solution in terms of traveling waves:

$$y(t,x) = y_r(t-\frac{x}{c}) + y_l(t+\frac{x}{c})$$ (2)

where $c = \sqrt{K/\varepsilon}$ denotes the wave propagation speed. Though each individual traveling wave is unobservable in the physical world, we use their constructs for modeling the physical behavior of the string, whose displacement is equal to the sum of the two traveling waves. Figures 3 to 10 show how a physical string's displacement, initially displaced to a triangular pulse, is equal to the sum of its left and right traveling waves at time $t=0$ . The blue arrows in Figure 3 show the points of displacement at time $t=0$ . Subsequent frames are shown, as the bottom plot of each frame corresponds to the physical displacement of the string. The top and bottom plots correspond to the right and left traveling waves, respectively. For a detailed derivation of the solution to the Wave Equation, we refer readers to
http://ccrma.stanford.edu/~jos/pasp/Traveling_Wave_Solution_I.html.

D'Alembert's solution to the wave equation serves as the theoretical foundation of which physical models for stringed instruments are based upon.