Brief Summary
This video explains the principle of superposition of waves, detailing constructive and destructive superposition with examples. It covers how waves overlap, the resulting amplitudes, and real-world examples like water surface waves. The video concludes with a summary of key concepts and an application exercise.
- Explains the principle of superposition of waves.
- Differentiates between constructive and destructive superposition.
- Provides examples and real-world applications.
Introduction to Superposition of Waves
The video introduces the concept of superposition of waves, which is the last part of chapter 2. It begins by reminding viewers about the vibrative state of points, explaining that points are in phase if they have the same vibratory state and out of phase if they have opposite states. Two points are in phase if the distance between them is a whole multiple of the wavelength (Lambda), and out of phase if the distance is an odd number divided by 2, multiplied by Lambda.
Principle of Superposition
The principle of superposition states that when two waves of the same nature (mechanical or electromagnetic) pass through the same point at the same time, they overlap. This overlap results in a new signal with a new amplitude at that instant. After superposition, the waves continue on their path with their original amplitudes. The video uses a diagram to illustrate two waves moving towards each other, overlapping, and then continuing with their original properties.
Constructive Superposition
Constructive superposition occurs when two signals have the same direction, meaning crests overlap with crests or troughs overlap with troughs. Before superposition, signals A and B have amplitudes y1 and Y2, respectively. During superposition, the amplitude of the resulting signal (A+B) is Y, calculated as y1 + Y2. After superposition, the amplitudes of signals A and B return to y1 and Y2. The video provides examples with specific amplitude values to illustrate the calculation.
Examples of Constructive Superposition
The video provides numerical examples to illustrate constructive superposition. In the first example, signal A has an amplitude of 2 mm and signal B has an amplitude of 3 mm. During superposition, the resulting amplitude is 2 + 3 = 5 mm. After superposition, each signal retains its original amplitude. A second example demonstrates the same principle with different amplitude values, reinforcing the concept.
Destructive Superposition
Destructive superposition occurs when two signals have opposite directions, resulting in the overlapping of a crest and a trough. Before superposition, signal A has an amplitude of absolute y1 and signal B has an amplitude of absolute Y2. During superposition, the amplitude of the resulting signal is the absolute value of y1 + Y2. After superposition, each signal continues with its original amplitude.
Examples of Destructive Superposition
The video presents examples to clarify destructive superposition. In one example, signal A (a crest) has an amplitude of 2 mm and signal B (a trough) has an amplitude of -3 mm. During superposition, the resulting amplitude is |2 + (-3)| = 1 mm. Another example shows two signals with equal amplitudes (2 mm) moving towards each other, resulting in a zero amplitude during superposition, illustrating complete cancellation.
Superposition of Waves
The video extends the concept of superposition to waves, showing how wave 1 and wave 2 superpose to form a resultant wave. In cases where the waves are in phase (crests overlap with crests, troughs with troughs), constructive superposition occurs, and the resulting wave's amplitude is the sum of the individual amplitudes. When waves are out of phase (crest overlaps with trough), destructive superposition occurs, potentially resulting in a zero amplitude if the waves have equal amplitudes.
Real-World Examples of Superposition
Superposition is illustrated with the example of waves on the surface of water produced by two sources, creating circular concentric waves. The red lines join points of maximum amplitude (constructive superposition), where crests or troughs meet. The black lines join points of minimum amplitude (destructive superposition), where a crest meets a trough. The figure demonstrates alternating constructive and destructive superposition.
Application Exercise
An application exercise is presented involving two disturbances propagating in opposite directions. The initial state of the disturbances is shown, with amplitudes of 2 meters and 1 meter, and a propagation speed of 1 meter per second. The task is to determine the state of the cord at T = 1, 2, and 3 seconds. The solution involves calculating the distance covered by each signal and determining whether constructive or destructive superposition occurs at each time interval.
Summary of Key Concepts
The video concludes with a summary of the key concepts: when two waves of the same nature pass through the same point at the same time, they overlap. Constructive superposition occurs when signals have the same direction, and the resulting amplitude is calculated as the sum of the individual amplitudes. Destructive superposition occurs when signals have opposite directions, and the resulting amplitude is calculated similarly.
