Example 8.

It was seen (p. 177) that this equation was derived from the original

where ${\displaystyle F}$ and ${\displaystyle f}$ were any arbitrary functions of ${\displaystyle t}$.

Another way of dealing with it is to transform it by a change of variables into

where ${\displaystyle u=x+at}$, and ${\displaystyle v=x-at}$, leading to the same general solution. If we consider a case in which ${\displaystyle F}$ vanishes, then we have simply

and this merely states that, at the time ${\displaystyle t=0}$, ${\displaystyle y}$ is a particular function of ${\displaystyle x}$, and may be looked upon as denoting that the curve of the relation of ${\displaystyle y}$ to ${\displaystyle x}$ has a particular shape. Then any change in the value of ${\displaystyle t}$ is equivalent simply to an alteration in the origin from which ${\displaystyle x}$ is reckoned. That is to say, it indicates that, the form of the function being conserved, it is propagated along the ${\displaystyle x}$ direction with a uniform velocity ${\displaystyle a}$; so that whatever the value of the ordinate ${\displaystyle y}$ at any particular time ${\displaystyle t_{0}}$ at any particular point ${\displaystyle x_{0}}$, the same value of ${\displaystyle y}$ will appear at the subsequent time ${\displaystyle t_{1}}$ at a point further along, the abscissa of which is ${\displaystyle x_{0}+a(t_{1}-t_{0})}$. In this case the simplified