What does this have to do with noise control? Well, along a one-dimensional path, the wave propagation has to satisfy these equations.
If f is an arbitrary function, a forward going pressure wave will take the form p(x,t)=f(x - c t) because (1.) it satisfies the wave equation, and (2.) the argument x - c t will remain a constant if and only if as t increases, x does also proportionally. Hence, as time increases, the function will produce the same value only for increasing x.
If you are in doubt, plug an example function, such as
, into the wave equation and do the derivatives.
A forward going pulse is shown in this ANIMATION.
Similarly if g is another arbitrary function, a backward going pressure wave will take the form p(x,t)=g(x + c t) because (1.) it also satisfies the wave equation, and (2.) the argument x + c t will remain a constant if and only if as t increases, x decreases proportionally. Here as time increases, the function will produce the same value for decreasing x.
A simple example of a function for a backward going wave
would be 
.
A backward going wave pulse is shown in this ANIMATION.