### Force Waves

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

(C.41) |

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

(C.42) |

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

(C.43) |

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.

**Next Section:**

Wave Impedance

**Previous Section:**

Spatial Derivatives