Let $m,k>0$, and $A,B$ be real. Then the differential equation

$$m{{d^{2}f}\over{dx^{2}}}+kf(x)=0$$

SHM

with initial conditions

$$f(0)=A,\quad f^{\prime}(0)=B$$

has solution

$$f(x)=A\cos\beta x+{{B}\over{\beta}}\sin\beta x$$

where $\beta=\sqrt{k/m}$, so $f$ is periodic with period $2\pi/\beta$. Consider a mass $m$ suspended on a spring with constant $k$ at time $x$ with displacement $f(x)$ from rest. The mass oscillates up and down.