Again the equation must be linearized and the variables split into mean and variable components. Rearranging and noting that ambient density changes with neither time nor position and that the condensation multiplied by the velocity is a very small number:  Euler's Force equation (conservation of momentum) is the last needed component. In one dimension the equation is:  where formula_33 represents the convective, substantial or material derivative, which is the derivative at a point moving with medium rather than at a fixed point. Linearizing the variables:  Rearranging and neglecting small terms, the resultant equation becomes the linearized one-dimensional Euler Equation:  Taking the time derivative of the continuity equation and the spatial derivative of the force equation results in:  Multiplying the first by formula_38, subtracting the two, and substituting the linearized equation of state,  The final result is  where formula_41 is the speed of propagation. Feynman provides a derivation of the wave equation for sound in three dimensions as  where formula_43 is the Laplace operator, formula_1 is the acoustic pressure (the local deviation from the ambient pressure), and where formula_6 is the speed of sound. A similar looking wave equation but for the vector field particle velocity is given by  In some situations, it is more convenient to solve the wave equation for an abstract scalar field velocity potential which has the form  and then derive the physical quantities particle velocity and acoustic pressure by the equations (or definition, in the case of particle velocity):  The following solutions are obtained by separation of variables in different coordinate systems. They are phasor solutions, that is they have an implicit time-dependence factor of formula_50 where formula_51 is the angular frequency. The explicit time dependence is given by Here formula_53 is the wave number.  where the asymptotic approximations to the Hankel functions, when formula_56, are  Depending on the chosen Fourier convention, one of these represents an outward travelling wave and the other a nonphysical inward travelling wave. The inward travelling solution wave is only nonphysical because of the singularity that occurs at r=0; inward travelling waves do exist.