### Wave Impedance

A concept of high practical utility is that of *wave impedance*,
defined for vibrating strings as force divided by velocity. As
derived in §C.7.2, the relevant *force* quantity in this case
is minus the string tension times the string slope:

(7.4) |

Physically, this can be regarded as the transverse force

*acting to the right*on the string in the vertical direction. (Only transverse vibration is being considered.) In other words, the vertical component of a negative string slope pulls ``up'' on the segment of string to the right, and ``up'' is the positive direction for displacement, velocity, and now force. The traveling-wave decomposition of the force into

*force waves*is thus given by (see §C.7.2 for a more detailed derivation)

^{7.2}

where we have defined the new notation for transverse velocity, and

*wave impedance*of the string for transverse waves. It is always real and positive for the ideal string. Three expressions for the wave impedance are

The wave impedance simply relates force and velocity traveling waves:

These relations may be called

*Ohm's law for traveling waves*. Thus, in a traveling wave, force is always

*in phase*with velocity (considering the minus sign in the left-going case to be associated with the direction of travel rather than a degrees phase shift between force and velocity).

The results of this section are derived in more detail in Appendix C. However, all we need in practice for now are the important Ohm's law relations for traveling-wave components given in Eq.(6.6).

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Velocity Waves at a Rigid Termination

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Sampled Traveling-Wave Solution