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When two waves travel along the same path, they interfere with each other, which can result in a new, combined wave known as the *resultant wave*. A key component of understanding wave interference is calculating the amplitude of this resultant wave.
To find this amplitude, we use the formula:

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

Here, \( A_1 \) and \( A_2 \) are the amplitudes of the two initial waves, and \( \phi \) is the phase difference between them. This equation takes into account both the individual amplitudes and the phase difference, allowing us to understand how much they will amplify or diminish each other's effects.
Using the values from the example, with \( A_1 = 4.60 \text{ mm} \), \( A_2 = 5.60 \text{ mm} \), and \( \phi = 0.80 \pi \text{ rad} \), you can calculate:

• \[ A_r = \sqrt{(4.60)^2 + (5.60)^2 + 2 \times 4.60 \times 5.60 \times \cos(0.80\pi)} \]

Knowing \( \cos(0.80\pi) = -0.309 \), we find the resultant amplitude to be approximately \( 6.05 \text{ mm} \). This shows how the initial waves combine to create a single wave with greater amplitude because of both constructive and destructive interference.

The phase angle is crucial in understanding how two waves combine. It describes the difference in the wave positions at a given point in time, relative to each other. This phase difference can determine whether the waves interfere constructively (amplifying the resultant wave) or destructively (diminishing it).
To calculate the resultant wave's phase angle \( \phi_r \) relative to the first wave, we apply:

• \[ \tan(\phi_r) = \frac{A_2 \sin(\phi)}{A_1 + A_2 \cos(\phi)} \]

In our example, this involves substituting \( A_1 = 4.60 \text{ mm} \), \( A_2 = 5.60 \text{ mm} \), \( \sin(0.80\pi) = 0.951 \), and \( \cos(0.80\pi) = -0.309 \):

• \[ \tan(\phi_r) = \frac{5.60 \times 0.951}{4.60 - 5.60 \times 0.309} \]
• \[ \tan(\phi_r) = \frac{5.32}{1.84} = 2.891 \]

The resultant phase angle, \( \phi_r \), can then be calculated using the inverse tangent function, yielding approximately \( 71.6^\circ \) or \( 1.25 \text{ rad} \).
Understanding phase angles helps visualize how wave phases align or misalign, leading to variations in amplitude.

When discussing wave interference, the *resultant wave* is the new wave formed by the combination of two or more waves traveling along the same medium. The attributes of this resultant wave, such as its amplitude and phase angle, indicate how the original waves interact.
For maximum amplitude in wave interference, the waves should interfere constructively. We achieve this by adjusting the phase angles of subsequent waves to match the resultant wave's phase angle. In our example, this means adding a third wave with an appropriate phase adjustment. To do this, the phase angle of the third wave \( \phi_3 \) should match the resultant phase angle, previously calculated as \( 1.25 \text{ rad} \).

• When the resultant wave's attributes are aligned perfectly with any additional waves, this ensures maximum amplitude due to constructive interference.
• This constructive interference is the principle behind technologies like noise-canceling headphones and radio broadcasting, where wave manipulation is crucial.

Thus, understanding how to develop and manipulate resultant waves allows for practical applications in various fields of science and engineering.