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Wave interference occurs when two or more waves overlap in a medium, causing them to combine and produce a new wave pattern. In simple terms, it's like when two musicians play the same note in sync, creating a richer sound. When dealing with two waves, such as the ones given by:

• \( y_1 = a \sin(\omega t - k x) \)
• \( y_2 = a \cos(\omega t - k x) \)

we need to add them algebraically to see how they interact.
In constructive interference, the waves are in phase, meaning their peaks and troughs align, resulting in a larger wave. Conversely, in destructive interference, they are out of phase, leading to a smaller wave or even cancellation. Here, the waves are sine and cosine, which relate but don't perfectly overlap. Understanding these interactions helps us find their resultant amplitude and how the wave behaves.
Superposition allows us to see the mix of their individual characteristics. Sprinkling in some math, the resultant wave is simply their addition, showcasing the beautiful math-based harmony of physics.

To solve problems involving superposition of waves, trigonometric identities become extremely helpful. These are rules that relate trigonometric functions like sine and cosine in ways that make complex equations more manageable.
In our problem, the trigonometric identity used is:

• \( \sin A + \cos A = \sqrt{2} \sin(A + \frac{\pi}{4}) \)

This identity transforms the separate sine and cosine terms into a single sine term with an adjusted argument.
By applying this identity to the problem, we simplify the expression:

• \( y = a[\sin(\omega t - k x) + \cos(\omega t - k x)] \)

becomes

• \( y = a \sqrt{2} \sin((\omega t - k x) + \frac{\pi}{4}) \)

The identity not only provides a straightforward path to solving equations but also highlights the close relationship between different trigonometric functions.

The resultant amplitude is the final height of the wave formed after interference occurs. It's determined by how the individual waves combine, influenced by both their amplitudes and phase differences.
In this instance, after using the trigonometric identity, we found that the waveform is

• \( y = a \sqrt{2} \sin((\omega t - k x) + \frac{\pi}{4}) \)

from waves \( y_1 \) and \( y_2 \).
This reveals that the resultant amplitude is \( a \sqrt{2} \), which reflects that the two original amplitudes, both \( a \), have combined to give a larger, single amplitude.
Because \( \sin A \) and \( \cos A \) have the same magnitude and are oriented such that their phases differ exactly by \( \frac{\pi}{2} \), they constructively interfere at 45 degrees. Hence, the maximum size of this combined wave is stretched by a factor of \( \sqrt{2} \).
This highlights how waves with similar frequencies, but differing in phase, can enhance each other when calculated appropriately.