Section 8.5: Superposition & Interference
When two or more waves overlap in space, the resulting displacement is the sum of individual displacements. This is the principle of superposition.
Depending on the relative phase of the waves, they can form:
- Constructive Interference: waves in phase reinforce each other. Resulting amplitude: \( A = A_1 + A_2 \)
- Destructive Interference: waves out of phase partially or completely cancel. Resulting amplitude: \( A = |A_1 - A_2| \)
Interference leads to patterns of maxima and minima, such as in sound beats, thin films, and double-slit experiments.
Example: Constructive Interference
Two waves of amplitude 3 cm meet in phase. Find the resultant amplitude.
\( A = A_1 + A_2 = 3 + 3 = 6 \text{ cm} \)
The resultant amplitude is 6 cm.
Example: Destructive Interference
Two waves of amplitude 5 cm and 3 cm meet completely out of phase. Find the resultant amplitude.
\( A = |A_1 - A_2| = |5 - 3| = 2 \text{ cm} \)
The resultant amplitude is 2 cm.
Practice Problems
- Two waves of amplitude 4 m meet in phase. Find the resultant amplitude.
- Two waves of amplitude 7 m and 2 m meet completely out of phase. Find the resultant amplitude.
- Explain the concept of beats using superposition.
- A double-slit experiment produces bright fringes 1.5 mm apart on a screen. Explain why this is constructive interference.
- Two sinusoidal waves traveling in the same medium are given by \( y_1 = 3 \sin(kx - \omega t) \), \( y_2 = 4 \sin(kx - \omega t + \pi) \). Find the resultant amplitude.