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Traveling waves are dynamic disturbances that move through a medium. Imagine throwing a stone into a pond. The ripples that spread outward are waves traveling away from the point of impact.
These waves carry energy and can be represented mathematically using equations that describe their sinusoidal nature, such as \(y(x, t) = y_m \sin(kx - \omega t)\) for a traveling wave. Here,

• \(y(x, t)\) is the wave function showing how the wave varies with position \(x\) and time \(t\).
• \(y_m\) is the amplitude, the maximum displacement of the wave.
• \(k\) is the wave number, linking to the wavelength, which is the distance over which the wave's shape repeats.
• \(\omega\) is the angular frequency, indicating how many cycles the wave goes through per unit of time.

Traveling waves are fundamental in understanding phenomena such as sound and light, illustrating how oscillations move through space over time.

When discussing wave interference, phase difference is a crucial concept. It tells us the shift in phase between two waves. Think of phase as a fraction of a wave cycle, like a clock's hands displaying time together with the sine wave's peaks and troughs.
A phase difference occurs when two waves with similar characteristics, like frequency and amplitude, overlap but start out of sync. This difference is usually measured in radians or degrees.
In the given problem, the phase difference \(\phi\) between the waves is \(\frac{\pi}{2}\) radians. This means one wave leads or lags the other by a quarter of a wavelength.

• If \(\phi\) is 0, waves are in phase, causing constructive interference where amplitudes add up maximally.
• If \(\phi\) is \(\pi\), waves are out of phase, potentially leading to destructive interference where they cancel each other out.
• Any other value of \(\phi\) results in partially constructive or destructive interference.

The specific phase difference influences the resultant wave's amplitude and can impact whether waves strengthen or weaken each other.

The sine addition formula is a mathematical tool used to add two sine functions together. It's particularly useful when dealing with the superposition of waves. The formula is expressed as:\[\sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)\]This tool allows us to transform two separate sinusoidal functions into a single expression representing their combined effect.
In the exercise, each wave is characterized by its own phase, leading to a superposition problem where we need to find the resultant amplitude.
When applying this formula:

• \(A\) and \(B\) are the phases of the individual waves.
• The \(\sin\) terms capture the wave's shape, while the \(\cos\) term helps reflect the phase difference's impact.
• For phases \(A = kx - \omega t\) and \(B = kx - \omega t + \frac{\pi}{2}\), the resultant wave combines these to yield a simplified single sine function.

The sine addition formula is integral in determining the amplitude modulation and the overall shape of the superposed wave and plays a key role in calculating the resultant wave's behavior.