Show that a stretched string is equivalent mathematically to an infinite number of uncoupled oscillators, described by the co-ordinates $$ q_{n}(t)=\sqrt{\frac{2}{l}} \int_{0}^{l} y(x, t) \sin \frac{n \pi x}{l} \mathrm{~d} x $$ Determine the amplitudes of the various normal modes in the motion described in Chapter 10, Problem 14. Why, physically, are the modes for even values of \(n\) not excited?

Fourier transformation is a mathematical concept that is widely used in various fields including string theory. In the context of a stretched string, this transformation method helps dissect the complex movement into simpler, more manageable sinusoidal waves. This is particularly useful because finding a solution to the motion of the entire string all at once can be daunting, if not impossible, for more complex motions.

By executing a Fourier transformation, we are considering the string's motion as a sum of vibrations at different frequencies. Each of these vibrations is a 'mode' of the string's oscillation, and their analysis is crucial to understanding the string's behavior. Imagine the string is vibrating in various patterns, some simple (like a single arch) and others more complex (like waves with several peaks and troughs). Fourier transformation allows us to view these patterns as a combination of simple oscillators moving in unison, making the analysis of the string's motion much simpler.

During the transformation process, the mathematical expressions for these modes are derived from integrals that combine the string's displacement at every point with sinusoidal functions. This approach is significant for both theoretical and practical reasons, offering insights into the mechanics of vibrating strings which could range from musical instruments to fundamental strings in theoretical physics.

In classical mechanics, oscillators represent physical systems that experience repeated motion. In the case of our stretched string, it behaves as an infinite set of uncoupled oscillators. Each point on the string can be seen as a separate oscillator, moving up and down as a result of forces restoring it to its equilibrium position.

When we say 'uncoupled,' we mean that the oscillators (the infinite points along a string's length) do not directly affect each other's motion. Despite this, they together form a wave pattern across the string due to the tension holding the string taut. The notion of breaking down the complex wave pattern into many oscillators is powerful—it means we can study one simple harmonic oscillator and then extrapolate our understanding to the entire string.

Oscillators are not just an abstract concept but play a role in everything from timekeeping in watches to electronic circuits. The relation between this concept and a stretched string helps bridge theoretical understanding with everyday phenomena.

The calculation of mode amplitudes is an essential step in understanding how a stretched string vibrates. Each mode of vibration carries a specific amplitude that determines the intensity or loudness of the motion at that frequency. The mathematical process to find these amplitudes uses the initial conditions of the string; basically, where and how fast each point on the string is moving at the start.

To find the amplitudes, we apply a Fourier transformation to these initial conditions. This gives us the Fourier coefficients, which directly relate to the mode amplitudes. These coefficients are essentially the 'ingredients' needed to recreate the initial wave form of the string when they are combined with the corresponding sinusoidal mode shapes.

By understanding the amplitudes of each mode, we gain crucial information about the behavior of the string when set into motion. This explains not just the dominant frequencies of its vibration, but also which frequencies will be almost non-existent due to having a very low amplitude.

In the study of stretched strings, the concept of eigenfunctions and their parity (even or odd) is critical. In mathematics, eigenfunctions corresponding to a specific problem are solutions that maintain their form after an operator is applied. In the case of our stretched string, these eigenfunctions are the sinusoidal modes that remain constant in form as the string vibrates.

Even eigenfunctions are symmetric around the midpoint of the string, while odd eigenfunctions are anti-symmetric. This symmetry property affects which modes are excited under certain initial conditions. As mentioned in the exercise solution, when the string's initial displacement and velocity are both odd functions, the modes for even values of n are not excited. This is because the integrals used to calculate the mode amplitudes (or the Fourier coefficients) for even n will sum up to zero, a direct result of the symmetry of the functions involved.

This distinction is not just a mathematical curiosity but has a real physical meaning. It tells us that if we pluck a string in a certain way, only certain patterns of vibration, or modes, will naturally resonate. Understanding this can help in designing musical instruments where string vibration is critical, or in any situation where controlling vibrational patterns is important.