D’Alembert’s solution

The one-dimensional wave equation in an infinite domain can be solved by a change of independent variables to reduce the wave equation that can be integrated to produce d Alembert s solution. Consider a one-dimensional wave equation. 

The earliest solution of the partial differential equation was given by D Alembert for the case of a vibrating string in 1750 [16]. At the same time, Bernoulli found a solution that was quite different from D Alembert s solution. Bernoulli s solution is based on the eigenfunction and is comparable with the Fourier series. 

General solutions (d Alembert s solution) predict two waves traveling at speed c in both the positive x-direction and the negative x-directiom... 

The physical meaning of this solution is that when M ) is increased or diminished by 21, the value of the function remains unaltered. Hence, when at is increased by 21, or, what is the same thing, when t is increased by 2JJa, the corresponding portions of the string will have the same displacement. In other words, the string performs at least one complete vibration in the time 21/a. We can show the same thing applies for 4Z, 6Z.. . . Hence, we conclude that d Alembert s equation represents a finite periodic motion, with a period of oscillation.. ... 

In linear differential equations, if a solution works, then it s a solution. The D Alembert solution to Equation B.8 is ... 

---