An oscillator with free period \(\tau\) is critically damped and subjected to a force with the 'saw-tooth' form $$ F(t)=c(t-n \tau) \quad \text { for } \quad\left(n-\frac{1}{2}\right) \tau

Understanding damped harmonic motion is essential when studying physical systems that oscillate with resistance, such as a critically damped oscillator. In general, harmonic motion refers to movement where the restoring force is proportional to the displacement, but in real-world scenarios, there's usually some form of resistance that slows down the oscillation.

This is where the concept of damping comes into play. An oscillator undergoing damped harmonic motion experiences a resistance force that is typically proportional to its velocity. The resistance reduces the energy of the system, consequently damping the oscillations over time. There are three types of damping: over-damped, under-damped, and critically damped.

In the critically damped case, the system returns to equilibrium without oscillating after being displaced. This specific type occurs when the damping coefficient is equal to the threshold value, which is determined by the mass and the spring constant of the system. Mathematically, the condition for critical damping is given by the equation \( b^2 = 4mk \), where \( b \) is the damping coefficient, \( m \) is the mass, and \( k \) is the spring constant. Critical damping is the fastest way to bring a system back to equilibrium without it overshooting the equilibrium position.

When solving problems involving critically damped oscillators, remember that the system does not exhibit sinusoidal motion like an under-damped oscillator. Instead, the motion will affect the relationship between force and displacement in a non-periodic manner, often requiring different analytical or computational techniques to resolve.

The Fourier series is a powerful mathematical tool used for analyzing and representing periodic functions as sums of simpler sinusoidal components. It can be especially useful in physics and engineering when dealing with periodic forces or signals.

For example, the saw-tooth wave in the exercise is a type of periodic force which can be broken down into a series of sine functions with the help of a Fourier series. The coefficients in a Fourier series, often noted as \( a_n \) and \( b_n \) for the sine and cosine terms respectively, inform us about the amplitude of each frequency component. Finding these coefficients allows us to reconstruct the original periodic function by summing its sinusoidal parts.

To calculate the Fourier coefficients for a specific periodic function like the saw-tooth force, we integrate the product of the function with sin and cos terms over one period of the function. It's through these coefficients that we find the amplitudes in various harmonics of the motion. Fourier series is invaluable when solving differential equations involving periodic forces, as it can transform a complex periodic function into a series of easier-to-manage terms.

The equation of motion is a fundamental concept that provides the relationship between the forces acting on a body and its motion. For an oscillator, such as the critically damped oscillator in our exercise, it describes how the oscillator moves under the influence of various forces.

In its simplest form, the equation of motion for an oscillator subject to an external force is given by Newton's second law, \( m\frac{d^2x}{dt^2} = F(t) - bx - kx \), where \( x(t) \) represents the displacement of the oscillator from its equilibrium position, \( m \) is its mass, \( b \) is the damping coefficient, \( k \) is the spring constant, and \( F(t) \) is the external force acting on the system.

In the context of our critically damped oscillator, we use the equation of motion to mathematically express how the saw-tooth force affects the oscillator's displacement. By introducing the Fourier series representation of the force into the equation, one can solve for the displacement as a sum of sinusoidal responses at different frequencies. Solving the equation of motion for its various components allows us to determine how the system behaves over time, giving us insight into amplitude ratios and other dynamic characteristics.