Referring to Fig.C.14, at an arbitrary point along the string, the vertical force applied at time to the portion of string to the left of position by the portion of string to the right of position is given by
assuming , as is assumed in the derivation of the wave equation. Similarly, the force applied by the portion to the left of position to the portion to the right is given by
These forces must cancel since a nonzero net force on a massless point would produce infinite acceleration. I.e., we must have at all times and positions .
Vertical force waves propagate along the string like any other transverse wave variable (since they are just slope waves multiplied by tension ). We may choose either or as the string force wave variable, one being the negative of the other. It turns out that to make the description for vibrating strings look the same as that for air columns, we have to pick , the one that acts to the right. This makes sense intuitively when one considers longitudinal pressure waves in an acoustic tube: a compression wave traveling to the right in the tube pushes the air in front of it and thus acts to the right. We therefore define the force wave variable to be
Note that a negative slope pulls up on the segment to the right. At this point, we have not yet considered a traveling-wave decomposition.