Elements of a Basic Bowed String Model

• Bow-string interaction

• Transverse waves on the string

• Losses at the bridge, finger, and along the string

This model supposes that the bow is applied to a single point $p_b$ in the string. When $v_b=v$ , bow and string stick together, otherwise they are sliding.

$$v = v_{o_n} + v_{i_n}$$

$$= v_{o_b} + v_{i_b}$$

The contribution of the reflected waves $v_{i_n}$ and $v_{i_b}$ are summed at the contact point:

$$v_h=v_{i_n}+v_{i_b}$$

Bow string interaction is represented by the following relations:

$$\left\{ \begin{array}{lclr} f & = & 2\; Z \left( v - v_h \right) & \\ f & = & \gamma \left( v - v_b \right) & \end{array}\right.$$

Once this coupling has been solved, the new outgoing waves $v_{o_n}$ and $v_{o_b}$ are calculated by the following equations:

$$\left\{ \begin{array}{lcl} v_{o_n} & = & v_{i_b} + \frac{f}{2Z} \\ v_{o_b} & = & v_{i_n} + \frac{f}{2Z} \end{array}\right.$$