The Wave Equation — Question 8

Question 8

A forcing term can be recovered from a failed homogeneous solution

A uniform string obeys $\mu y_{tt}=F y_{xx}+p(x,t)$, where $p$ is an externally applied transverse force per unit length. A prescribed motion is $y=A\cos(kx)\cos\Omega t$.

1. Find the required force density.

2. For $\Omega=vk$ with $v^2=F/\mu$, what forcing is required in the interior? Why might an experiment still require a driver?

Question 8 - Solution

A forcing term can be recovered from a failed homogeneous solution

Method and mathematical working

Move the inertial and tension terms to the side defining the applied force per length. Evaluate both second derivatives of the prescribed standing profile.$$\begin{aligned}p&=\mu y_{tt}-F y_{xx},\\y_{tt}&=-\Omega^2y,\qquad y_{xx}=-k^2y,\\p&=(Fk^2-\mu\Omega^2)y.\\[4pt]p&=\mu(v^2k^2-\Omega^2)y,\qquad\Omega=vk\Rightarrow p=0.\end{aligned}$$

(a) Calculation and conclusion

$y_{tt}=-\Omega^2y$ and $y_{xx}=-k^2y$, so $p=\mu y_{tt}-F y_{xx}=(Fk^2-\mu\Omega^2)y$. Its units are force per unit length.

(b) Interpretation and limits

At $\Omega=vk$, the interior forcing vanishes: this is a free solution if boundaries and initial conditions are compatible. A real system can still need a driver to replace dissipative losses or establish the motion. The ideal homogeneous equation contains no such losses.

Physical check

When $\Omega^2=Fk^2/\mu$, the required interior forcing vanishes. Endpoint compatibility and the initial state still determine whether that unforced motion is physically realized.