The two waves \(\eta_{1}=A_{1} \cos (\kappa x-\omega t)\) and \(\eta_{2}=A_{2} \cos (\kappa x+\omega t)\) travel together on a stretched string. Show how the wave amplitude ratio \(A_{1} / A_{2}\) can be found from a measurement of the so-called standing-wave ratio \(S\) of the resulting wave pattern, which is defined as the ratio of the maximum amplitude to the minimum amplitude of the envelope of this pattern. Note that \(S\) and the position and amplitude of the minima (or maxima) wave disturbance uniquely determine the two traveling waves that give rise to the standing-wave pattern.

When discussing waves, amplitude is a crucial term. The amplitude refers to the highest displacement from the rest position in a wave. This measurement is directly linked to the energy carried by the wave; the greater the amplitude, the more energy it transports. In the context of the provided exercise, two wave amplitudes, represented by A₁ and A₂, play a significant role. When two waves combine, as they do on a string, their individual amplitudes contribute to the overall wave dynamics—this is essential for understanding the standing-wave ratio (S). One way to visualize amplitude is by picturing a calm sea (low amplitude) versus towering ocean waves (high amplitude); the latter obviously contain more energy and have a higher potential to impact the shore (or, in our case, the string's displacement).

The concept of a standing wave pattern is fascinating as it appears to stand still, though it results from the interference of two traveling waves moving in opposite directions. Such patterns are commonly seen in musical instruments like guitars and violins. When two waves with the same frequency and wavelength meet while traveling in opposite directions, their crests and troughs can align at certain points, resulting in points of no displacement called nodes, and points of maximum displacement, or antinodes. This gives rise to the standing wave pattern with its characteristic stationary appearance, despite being the product of continuous motion. The exercise we're examining investigates how to derive quantitative information about the wave amplitudes based on the standing-wave ratio—specifically the amplitude ratio A₁/A₂—which can be observed and measured from the pattern's envelope, or the outer limit of the wave's oscillation.

Trigonometric identities are like the swiss army knives of mathematics, offering various ways to simplify and solve complex problems. The sum-to-product identities, in particular, are helpful when dealing with wave equations in physics. They allow the conversion of a sum or difference of two trigonometric functions into a product of trigonometric functions, making the equations more manageable. This transformation is the key step in evaluating the wave patterns that result from the superposition of two waves, as seen in the exercise. By applying a sum-to-product identity, we can move from a complex expression involving the sum of two cosines to a simpler, more insightful formula that represents the standing wave's envelope. Understanding these identities is essential for students tackling problems involving wave interactions, as they lay the groundwork for elucidating the resultant wave's amplitude and pattern.

Wave superposition is a central concept in wave mechanics, describing how multiple waves can coexist and overlap, combining to form a new wave pattern. The principle of superposition states that when two or more waves meet, the total displacement at any point is the sum of the displacements due to each wave individually. This principle is illustrated vividly when two waves traverse the same medium, such as a string. If the waves have compatible frequencies and wavelengths, they can interfere constructively or destructively at various points—constructively where their amplitudes reinforce each other, and destructively where they cancel each other out. The exercise sheds light on how this concept is not only theoretical but can be quantified. By applying superposition, we can mathematically determine the amplitudes of the individual waves that construct the observed standing wave pattern on the string, making the abstract concrete.