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For $f(x)=|x|$ and positive amplitude $A$, the averaged amplitude for position-dependent damping givesThe average is $(4/(2\pi )){\int }_0^{\pi /2}\cos \psi {\sin }^2\psi d\psi =2/(3\pi )$. Thus the absolute-value damping amplitude law isand ${\theta }_T=0$ as in the other cases. Consequently,The dissipation makes the leading envelope decrease algebraically in slow time. The absolute value is Lipschitz continuous, so the first-order averaging argument remains valid despite its corner at zero.