Force Waves

• Vertical force acting to the left is
  $$f_l(t,x) = K\sin(\theta) \approx K\tan(\theta) = K\, y'(t,x)$$

• Opposing force, acting to the right, is
  $$f_r(t,x) = - K\sin(\theta) \approx - K\, y'(t,x)$$
  (Note that a negative slope pulls ``up'' on the segment to the right)

• These forces must cancel since a nonzero net force on a massless point would produce infinite acceleration

To unify vibrating strings with acoustic tubes, we choose the force which acts to the right as our force wave variable:

$$f(t,x) \mathrel{\stackrel{\mathrm{\Delta}}{=}}f_r(t,x) = \zbox{-Ky'(t,x)}$$

• Analogous to longitudinal pressure in acoustic tubes
• We have
  $$f(t,x) = \frac{K}{c} \left[{\dot y}_r(t-x/c) - {\dot y}_l(t+x/c) \right]$$

• Force waves are thus proportional to velocity waves
• Proportionality constant is called the wave impedance
  (or characteristic impedance) of the string:
  $$R\mathrel{\stackrel{\mathrm{\Delta}}{=}}\sqrt{K\epsilon } = \frac{K}{c} = \epsilon c$$

• Wave impedance = geometric mean of spring stiffness and inertial mass
• Traveling force-wave components:
  $$\begin{array}{rcrl} f^{{+}}(n)&=&&R\,v^{+}(n) \\ f^{{-}}(n)&=&-&R\,v^{-}(n) \end{array}$$

For acoustic tubes, we have

$$\begin{array}{rcrl} p^+(n) &=& &R_{\hbox{\tiny T}}\, u^{+}(n) \\ p^-(n) &=& -&R_{\hbox{\tiny T}}\, u^{-}(n) \end{array}$$

where

• $p^+(n) =$ right-going longitudinal pressure
• $p^-(n) =$ left-going longitudinal pressure
• $u^\pm (n) =$ left and right-going volume-velocity waves
• wave impedance is
  $$\zbox{R_{\hbox{\tiny T}}= \frac{\rho c}{A}} \qquad\hbox{(Acoustic Tubes)}$$
  where
  • $\rho =$ mass per unit volume of air
  • $c =$ sound speed in air
  • $A =$ cross-sectional area of tube
• For particle velocity, wave impedance $= R_0=\rho c$
• Particle velocity is appropriate in open air, while volume velocity is used for acoustic tubes