Origin of the wave equation: vibrating springs

The wave equation is a partial differential equation that describes the propagation of waves through a medium, such as mechanical waves (such as sound) through air or water, or electromagnetic waves (such as visible light or radio waves) through vacuum or glass.

For a function $f(t)$ with a single variable $t$ representing time, the zero-dimensional wave equation has the form

This equation describes the horizontal position of a point mass (with $m=1$) attached to a (idealized and frictionless) spring, where the force (second-derivative) is proportional and opposite to the displacement from equilibrium. For a spring with spring constant $k$ (representing the stiffness), the equation becomes (we have normalized mass to 1) $$f''(t) = -k f(t)$$

The general solution to this equation is a sinusoidal function, and has the form $$f(t) = A\cos(\omega (t-\phi))$$ where $A$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ is the phase shift. The angular frequency $\omega$ is related to the spring constant $k$ by $\omega = \sqrt{k}$. The other two parameters are related to the initial position and velocity of the mass. For example, if the mass is initially at rest at position $x_0$, then $A = x_0$ and $\phi = 0$, giving $f(t) = x_0\cos(\sqrt{k}t)$. If the mass is initially moving with velocity $v_0$, then $A = \sqrt{x_0^2 + v_0^2}$ and $\phi = \arctan(v_0/x_0)$.

In the applet below, the point mass is represented by a black square and can be dragged horizontally to change its position. Its equilibrium position is depicted by a dashed line, and the force acting on the mass is shown as a red arrow. By clicking the play button, you can observe how the mass moves over time. To its right, the position $f(t)$ is graphed through time. Two sliders are present which can be used to adjust the stiffness coefficient $k$, and to add friction to the spring.

Question:Adding friction changes the solution to the equation. How do you think it should affect the graph of $f(t)$? And how do you think it should affect the general solution to the equation?

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