Draw a sketch like that of Problem 4 but this time draw the...

Draw a sketch like that of Problem 4 but this time draw the third wave $180^{\circ}$ out of phase with the others. (a) What is the amplitude of the sum of these waves? (b) What is the intensity for the four waves together? (c) Consider the case for the first three waves in phase and the fourth wave $180^{\circ}$ out of phase. What is the amplitude for the sum of these waves? (d) What is the intensity of the wave?

Draw a sketch like that of Problem 4 but this time draw the third wave $180^{\circ}$ out of phase with the others.
(a) What is the amplitude of the sum of these waves?
(b) What is the intensity for the four waves together?
(c) Consider the case for the first three waves in phase and the fourth wave $180^{\circ}$ out of phase. What is the amplitude for the sum of these waves? (d) What is the intensity of the wave?

Key Concepts

Amplitude-Intensity Relationship

The intensity of a wave is related to the square of its amplitude. This relationship is crucial when analyzing wave phenomena because any change in the resultant amplitude due to interference directly affects the intensity observed.

Constructive and Destructive Interference

Constructive interference occurs when waves are in phase, resulting in an increased amplitude, while destructive interference happens when waves are out of phase, which can reduce or even completely cancel the resultant amplitude. This concept is key in predicting the outcomes of combining multiple waves with varying phase relationships.

Vector Addition of Waves

Since waves have both magnitude (amplitude) and phase (direction in the context of a phasor diagram), adding them often involves treating amplitudes as vectors. This method of summing amplitudes aids in accurately determining the resultant wave when phases differ.

Superposition Principle

This principle states that when two or more waves travel through the same medium, the net displacement at any point is the sum of the displacements due to the individual waves. It is essential for understanding how multiple waves combine, whether they interfere constructively or destructively.

Phase Difference

A phase difference between waves describes how much one wave is shifted relative to another. A phase shift of 180° (or ? radians) means that the waves are exactly out of phase, leading to situations where peaks align with troughs, potentially causing cancellation effects.

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