The 1D Wave Equation

• The wave equation provides an analytic description of wave motion over time and through a spatial medium.

• This mathematical equation can subsequently be used to evaluate and model wave motion along the string and at boundaries.

• In analyzing the one-dimensional (1D) string section illustrated in Fig. 1 below, we make the following assumptions:
  • The mass per unit length of the string is constant and the string is perfectly elastic (there is no resistance to bending).
  • The tension caused by stretching the string before fixing it at its endpoints is so large that gravitational forces on the string are negligible.
  • The string moves only in the transverse direction and these deflections are small in magnitude.

  Figure 1: A short section of a stretched, deformed string.

  

• The mass of the short string section (length $\Delta x$) is $\epsilon \Delta x$, where $\epsilon$ is the mass per unit length of the string.

• Since there is no horizontal motion, the two horizontal components of tension must be constant: $T_{1} \cos \theta_{1} = T_{2} \cos \theta_{2} = T$.

• The net vertical force on the section is $T_{2} \sin \theta_{2} - T_{1} \sin \theta_{1}$.

• By Newton's Second Law: $T_{2} \sin \theta_{2} - T_{1} \sin \theta_{1} = \epsilon \Delta x \frac{\partial^{2}y}{\partial t^{2}}$.

• Making use of the horizontal tension components, we obtain:
    $$\frac{T_{2} \sin \theta_{2}}{T_{2} \cos \theta_{2}} - \frac{T_{1}... ... \theta_{1} = \frac{\epsilon \Delta x}{T} \frac{\partial^{2}y}{\partial t^{2}}$$ (1)

• The expressions $\tan \theta_{2}$ and $\tan \theta_{1}$ are the slopes of the string at the points $x$ and $x + \Delta x$:
  $$\tan \theta_{1} = \left( \frac{\partial y}{\partial x}\right)_x \... ...n} \tan \theta_{2} = \left( \frac{\partial y}{\partial x}\right)_{x+\Delta x}.$$

• Making these substitutions in Eq. 1 and dividing by $\Delta x$,
  $$\frac{1}{\Delta x} \left[\left( \frac{\partial y}{\partial x}\rig... ...x}\right)_x \right] = \frac{\epsilon}{T} \frac{\partial^{2}y}{\partial t^{2}}.$$

• Letting $\Delta x$ approach zero, we obtain the linear partial differential equation
  $$\frac{\partial^{2} y}{\partial t^{2}} = \frac{T}{\epsilon} \frac{\partial^{2} y}{\partial x^{2}} = c^{2} \frac{\partial^{2} y}{\partial x^{2}},$$
  where $c = \sqrt{T/\epsilon}$ is the speed of wave motion on the string. This is the one-dimensional wave equation that describes small amplitude, lossless transverse waves on a stretched string.

• Another way of interpreting the 1D wave equation is that the vertical acceleration of the string at a given location is proportional to its curvature. For the string section illustrated in Fig. 1, the curvature is negative (the slope at $x + \Delta x$ is less than the slope at $x$), and thus the resulting acceleration of this infinitesimal section will be negative (or in the negative $y$ direction).