The Principle of Superposition for Waves — Question 10

Question 10

Adding two driven solutions also adds their applied forces

Let $\mathcal L[y]=\mu y_{tt}-Fy_{xx}$. A field $y_1$ satisfies $\mathcal L[y_1]=p_1$ and $y_2$ satisfies $\mathcal L[y_2]=p_2$. Boundary conditions are homogeneous and linear.

1. Find the force density required by $y_1+y_2$.

2. If two different solutions each have the same applied load $p$, what equation does their difference satisfy? Explain the role of initial conditions.

Question 10 - Solution

Adding two driven solutions also adds their applied forces

Method and mathematical working

Apply linearity to the entire forced equation, including its right-hand side. Addition and subtraction give different driving terms.$$\begin{aligned}\mathcal L[y_1+y_2]&=p_1+p_2,\\\mathcal L[y_1-y_2]&=p_1-p_2.\\[4pt]p_1=p_2=p&\Rightarrow\mathcal L[y_1+y_2]=2p,\qquad\mathcal L[y_1-y_2]=0.\end{aligned}$$

(a) Calculation and conclusion

Linearity gives $\mathcal L[y_1+y_2]=p_1+p_2$. In particular, adding two responses to the same nonzero force generally doubles that force; it does not produce another response to the original force.

(b) Interpretation and limits

For equal loads, $\mathcal L[y_1-y_2]=0$. Their difference is a free solution determined by differences in initial displacement and velocity. With identical compatible initial and boundary data, uniqueness eliminates that difference. Superposition must track sources and data as well as fields.

Physical check

When the two drives are identical, their difference obeys the homogeneous equation. It vanishes only if the relevant initial and boundary differences also vanish; equal forcing alone is insufficient.