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When two or more waves traverse the same space, they interact with each other. This interaction is known as wave interference, which can be constructive, destructive, or a complex pattern of both. Constructive interference occurs when wave crests (or troughs) from different waves align, resulting in a wave of greater amplitude. Conversely, destructive interference happens when wave crests of one wave align with the troughs of another, reducing the wave's overall amplitude.

In the given exercise, we observe the superposition of two different waves on a string. When they meet, their respective displacements combine to form the resultant wave at any point and time. This principle is imperative in fields ranging from acoustics and optics to quantum mechanics. Understanding how to add wave functions algebraically is essential for predicting the combined influence of multiple waves on a medium.

Trigonometric functions like sin and cos are fundamental in describing wave motion in physics. These mathematical tools allow us to model the oscillatory behaviour of waves as they travel through space and time. The general wave function often takes the form \(y = A \times \text{trig}(kx - \theta t + \text{phase})\), where \(A\) represents the amplitude, \(k\) the wave number, \(\theta\) the angular frequency, and \(\text{phase}\) is the phase constant. Specifically, the cosine function is used when a wave begins at its maximum displacement, while sine is employed when it starts from the equilibrium position.

In our exercise, these trigonometric functions are used within wave functions to describe the position of points on a vibrating string over time, allowing us to calculate the superposition at given instances.

A wave function in physics describes the characteristics of a wave at any given point and time. It succinctly represents the wave's shape, amplitude, wavelength, frequency, and phase. The wave function equations provided in the exercise, \(y_{1} = 3.0 \cos (4.0 x - 1.6 t)\) and \(y_{2} = 4.0 \sin (5.0 x - 2.0 t)\), each give a mathematical description of individual waves on the string. The cos and sin functions vary between -1 and 1, perfect for demonstrating the oscillatory nature of waves.

By adding these functions together, we obtain the superposition wave function. This effectively allows us to predict the motion and interaction of waves, forming the backbone of much of modern physics. Applications of wave functions include determining sound volume at a point due to speakers in various locations or predicting the patterns seen in wave interference experiments in the lab.