The Wave Equation — Question 1

Question 1

The wave equation follows from a difference of transverse tensions

A uniform string has tension $F$, density $\mu$ and small transverse slope $y_x$. Consider a short horizontal interval $dx$, with no external transverse load.

1. Derive the linear equation of motion and identify its wave speed.

2. Explain why the net force depends on curvature rather than the slope itself.

Question 1 - Solution

The wave equation follows from a difference of transverse tensions

Method and mathematical working

Apply Newton's law to a short element, keeping only first-order slopes. Divide by its length and take the limit to obtain a differential equation.$$\begin{aligned}F[y_x(x+dx)-y_x(x)]&=\mu\,dx\,y_{tt},\\F y_{xx}&=\mu y_{tt},\qquad v^2=F/\mu.\\[4pt][F/\mu]&=\frac{\mathrm{kg\,m/s^2}}{\mathrm{kg/m}}=\mathrm{m^2/s^2},\qquad y_{xx}=0\Rightarrow y_{tt}=0.\end{aligned}$$

(a) Calculation and conclusion

The right tension contributes approximately $F y_x(x+dx)$ upward and the left contributes $-F y_x(x)$. Their sum is $F y_{xx}dx$. Newton's law gives $\mu dx\,y_{tt}=F y_{xx}dx$, or $y_{tt}=v^2y_{xx}$ with $v=\sqrt{F/\mu}$.

(b) Interpretation and limits

A constant slope gives equal transverse tension components at the two ends, so they cancel. Only their spatial change produces a net force. The derivation assumes small slopes, constant tension and density, and negligible bending rigidity.

Physical check

A straight segment with uniform slope has equal transverse endpoint components and no net transverse force. Curvature, rather than displacement or slope alone, causes acceleration.