The Principle of Superposition for Waves — Question 9

Question 9

A nonlinear restoring term prevents unrestricted superposition

Consider the model $y_{tt}-v^2y_{xx}+\gamma y^2=0$, with $\gamma\ne0$. Two functions $y_1,y_2$ each solve it separately.

1. Find the residual obtained by substituting $y_1+y_2$.

2. State when this residual vanishes and explain why small amplitude can make superposition an approximation rather than an exact rule.

Question 9 - Solution

A nonlinear restoring term prevents unrestricted superposition

Method and mathematical working

Substitute the sum into the full nonlinear equation. The individual residuals vanish, leaving the mixed term from the square.$$\begin{aligned}\gamma(y_1+y_2)^2&=\gamma y_1^2+\gamma y_2^2+2\gamma y_1y_2,\\\mathcal N[y_1+y_2]&=2\gamma y_1y_2.\\[4pt]y_1,y_2=O(\epsilon)&\Rightarrow2\gamma y_1y_2=O(\epsilon^2),\qquad\epsilon\to0:\ \text{linear limit}.\end{aligned}$$

(a) Calculation and conclusion

The linear derivative terms add, but $\gamma(y_1+y_2)^2$ includes $2\gamma y_1y_2$. Subtracting the two individual equations leaves residual $2\gamma y_1y_2$.

(b) Interpretation and limits

The residual vanishes where one field is zero, or in special cases with zero product. In general overlap it is nonzero. Scaling both amplitudes by a small factor makes the nonlinear residual quadratic in that factor, compared with linear terms, allowing a controlled leading-order approximation when the model scales justify it.

Physical check

The residual is second order in small displacement amplitudes. Neglecting it can justify an approximate linear theory, but exact superposition requires the nonlinear coupling to vanish.