91Ó°ÊÓ

The Superposition Principle is a fundamental concept in wave mechanics. It states that when two or more waves traverse the same space, their displacements add together algebraically at every point. This principle helps us understand complex wave interactions and solve real-world problems by breaking them into simpler components.
For instance, when waves travel along the same string, the resultant wave can be accurately predicted by adding the individual waves' equations. It’s like combining overlapping pictures to create a composite image. In our exercise, the two given waves are described by the equations \( y_1(x, t) \) and \( y_2(x, t) \).
Their resultant wave, \( y(x, t) \), is created by adding these two:

• \( y(x, t) = y_1(x, t) + y_2(x, t) \)

This concept explains how waves interfere with each other, leading to either an increase or decrease in the overall amplitude depending on their relative phases.

Resultant Amplitude is the combined amplitude of multiple waves, measuring the wave's maximum displacement. Calculating this involves taking into account both amplitude magnitudes and their relative phase angles.
To solve for the resultant amplitude, the formula used is:

• \( A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos \phi} \)

Here, \( A_1 \) and \( A_2 \) represent individual wave amplitudes, and \( \phi \) is the phase difference. In our exercise, these values are plugged in to obtain:

• \( A_1 = 4.00 \text{ mm} \)
• \( A_2 = 6.20 \text{ mm} \)
• \( \phi = 0.60 \pi \)

This results in a calculated amplitude of about 6.26 mm, combining the influences of both contributing waves.

The Phase Angle is a critical concept in understanding how waves interfere with each other. It represents the offset between wave cycles and governs the wave interference pattern.
To determine the resultant phase angle, we consider the phase angles of individual waves and their interactions, which can be calculated using the formula:

• \( \tan \theta = \frac{A_2 \sin \phi}{A_1 + A_2 \cos \phi} \)

For the given exercise, after substituting values:

• \( \tan \theta = \frac{6.20 \times 0.951}{4.00 + 6.20 \times (-0.309)} \)

This calculates the resultant phase angle \( \theta \) as approximately 70.83 degrees. This angle is pivotal for predicting how the wave behaves under interference conditions.

Constructive Interference occurs when two or more waves align such that their crests and troughs coincide, resulting in a higher amplitude. This is the 'adding up' of waves that leads to an enhanced wave effect.
Maximizing amplitude through constructive interference involves tuning phase angles to ensure wave peaks line up. With a third wave introduced to the scenario described in the exercise, it can achieve maximized amplitude—a key outcome of constructive interference—by setting its phase angle to align with the resultant wave phase angle.

• This means setting its phase angle to approximately \( 70.83^{\circ} \) to match the previously calculated angle for maximum collective amplitude.

Thus, by aligning wave phases, we exploit constructive interference to amplify the resulting wave efficiently.