Longitudinal String Waves

Consider longitudinal force and velocity waves in a string:

$$\begin{array}{rcrl} f^{{+}}_i(t)&=&&R_iv^{+}_i(t) \\ f^{{-}}_i(t)&=&-&R_iv^{-}_i(t) \end{array}$$

where

• Force-variable $f=f^{{+}}+f^{{-}}$ now denotes
  longitudinal stress = longitudinal compression pressure (force/area) -- a force density
• Wave impedance is now $R_i=\sqrt{E\rho}$
• $\rho =$ mass density
• $E =$ Young's modulus
  (stress/strain = compressionPressure/relativeDisplacement)