The Principle of Superposition for Waves — Question 1

Question 1

Linearity applies to displacement before it applies to any energy calculation

Two solutions $y_1,y_2$ satisfy $y_{tt}=v^2y_{xx}$ on the same uniform string. Let $y=ay_1+by_2$, where $a,b$ are constants.

1. Prove that $y$ satisfies the same equation.

2. Why does this proof not establish that the energy density equals $a^2u_1+b^2u_2$ at every point?

Question 1 - Solution

Linearity applies to displacement before it applies to any energy calculation

Method and mathematical working

Apply the linear differential operator separately to each term in the proposed combination. Then expand the quadratic energy to see why a different rule applies to energy.$$\begin{aligned}\mathcal L[ay_1+by_2]&=a\mathcal L[y_1]+b\mathcal L[y_2]=0,\\u[y_1+y_2]&=u[y_1]+u[y_2]+\mu y_{1t}y_{2t}+F y_{1x}y_{2x}.\\[4pt](ay_1+by_2)_{tt}-v^2(ay_1+by_2)_{xx}&=a(0)+b(0)=0.\end{aligned}$$

(a) Calculation and conclusion

Linear differentiation gives $y_{tt}-v^2y_{xx}=a(y_{1,tt}-v^2y_{1,xx})+b(y_{2,tt}-v^2y_{2,xx})=0$. Both fields must obey the same equation and domain assumptions.

(b) Interpretation and limits

Energy contains squared derivatives. Expanding gives cross terms $ab[\mu y_{1,t}y_{2,t}+F y_{1,x}y_{2,x}]$, which need not vanish locally. Superposition is a statement about the displacement equation, not automatic addition of every quadratic observable.

Physical check

Superposition of displacement follows from a linear equation of motion. Additivity of energy requires the cross terms to vanish or cancel after the appropriate integration; it is a separate question.