text: The homogeneous wave equation (i.e. Eq. () when  is zero) has solutions  for travelling waves of permanent form propagating in either the negative or positive -direction. For the inhomogeneous case, considering waves propagating in the positive -direction, Green proposes an approximate solution:
  Then
  Now the left-hand side of Eq. () becomes:
  So the proposed solution in Eq. () satisfies Eq. (), and thus also Eq. () apart from the above two terms proportional to  and , with  The error in the solution can be made of order  provided
  This has the solution:
  Using Eq. () and the transformation from  to , the approximate solution for the surface elevation  is
  where the constant  has been set to one, without loss of generality. Waves travelling in the negative -direction have the minus sign in the argument of function  reversed to a plus sign. Since the theory is linear, solutions can be added because of the superposition principle.