Force Wave Variables

Force Equation:

$$f_m(t) + f_{1m}(t) + f_{2m}(t) = 0,$$

Traveling-Wave Decomposition of String Force:

$$\begin{eqnarray*} f_1(t,x)&=&f^{{+}}_1(t-x/c)+f^{{-}}_1(t+x/c)\\ f_2(t,x)&=&f^{{+}}_2(t-x/c)+f^{{-}}_2(t+x/c) \end{eqnarray*}$$

String Force:

$$f(t,x) \isdef -Ky'(t,x)$$

where

• $K =$ string tension
• $y'=$ string slope

Note that string force pulls up to the right.
That is, a string segment with negative slope pulls ``up'' to the right and ``down'' to the left:

In the present problem, our force equation

$$f_m(t) + f_{1m}(t) + f_{2m}(t) = 0,$$

becomes, in terms of mass inertial force and string forces:

$$m\dot v(t) + K\,y'_1(t,0) - K\, y'_2(t,0) = 0$$

or, using string force-wave notation:

$$\zbox{f_m(t) - f_1(t) + f_2(t) = 0}$$

These force relations can be checked individually:

• For string 1,
  $$m\dot v(t) + K\,y'_1(t,0) = 0$$
  $\implies$ positive slope on left accelerates mass down

• For string 2,
  $$m\dot v(t) - K\,y'_2(t,0) = 0$$
  $\implies$ positive slope on right accelerates mass up