Transverse Wave Equation: Ideal String

Wave Equation

$$\zbox{ Ky''= \epsilon {\ddot y}}$$

$$\begin{array}{rclrcl} K& \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \hbox{string tension} & \qquad y & \mathrel{\stackrel{\mathrm{\Delta}}{=}}& y(t,x) \nonumber \\ \epsilon & \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \hbox{linear mass density} & {\dot y}& \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \frac{\partial}{\partial t}y(t,x) \nonumber \\ y & \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \hbox{string displacement} & y'& \mathrel{\stackrel{\mathrm{\Delta}}{=}}& \frac{\partial}{\partial x}y(t,x) \nonumber \end{array}$$

Newton's second law

$$\zbox{\hbox{Force} = \hbox{Mass} \times \hbox{Acceleration}}$$

Assumptions

• Lossless
• Linear
• Flexible (no ``Stiffness'')
• Slope $y'(t,x) \ll 1$