two stones are dropped simultaneously into a calm pool of water. The crests of the resulting waves form equally spaced concentric cicles, as shown in the figures. The waves interact with each other to create certain interference patterns.

explain why the red dots lie on an ellipse
explain why the blue dots lie on a hyperbola

The red dots lie on an ellipse because the two stones dropped into the pool of water create two circular waves that interfere with each other. The interference of the two waves creates a pattern of constructive and destructive interference, which results in the red dots forming an ellipse.

The blue dots lie on a hyperbola because the two stones dropped into the pool of water create two circular waves that interfere with each other. The interference of the two waves creates a pattern of constructive and destructive interference, which results in the blue dots forming a hyperbola. The hyperbola is formed because the constructive interference of the two waves creates a peak, while the destructive interference of the two waves creates a trough.

To understand why the red dots lie on an ellipse while the blue dots lie on a hyperbola, we need to explore the concept of interference patterns created by waves.

When two waves are introduced into water simultaneously, they create a phenomenon called wave interference. This occurs because each wave's disturbance pattern combines with the disturbance pattern of the other wave, resulting in variations in the amplitude (height) of the resulting wave.

Now, let's break down the reasons for the red dots lying on an ellipse:

1. Constructive Interference: In some regions where the crests of the waves overlap, they reinforce each other. This is called constructive interference. Constructive interference creates areas of higher amplitude, resulting in locations where the water's surface appears higher than the surrounding areas.

2. Destructive Interference: In other regions where a crest and trough overlap, they cancel each other out. This is called destructive interference. Destructive interference creates areas of lower amplitude, resulting in locations where the water's surface appears lower than the surrounding areas.

3. Superposition: When the waves are introduced simultaneously, their individual effects are combined through a process called superposition. Superposition is the principle that states that when waves meet, the resulting displacement at any point is the sum of the individual displacements caused by each wave.

Now, let's move on to explaining why the blue dots lie on a hyperbola:

1. Hyperbolic Property: In interference patterns, the blue dots, which represent areas of constructive interference, exhibit a property known as hyperbolic shape.

2. Interference Pattern: As the waves interact with each other, the areas of constructive interference form regions with higher amplitude. These regions represent wave crests or the highest points of interference.

3. Hyperbolic Shape: The specific geometry of wave interference leads to the pattern of the highest points being arranged in a hyperbolic shape. This shape is characteristic of constructive interference in wave interference patterns.

In summary, the red dots lie on an ellipse because they represent areas of constructive and destructive interference, resulting in variations in wave amplitude. On the other hand, the blue dots lie on a hyperbola because they represent areas of constructive interference and exhibit a distinct hyperbolic shape.

The red dots in the figure lie on an ellipse because the waves created by the two stones exhibit constructive and destructive interference.

When two waves meet in phase (crest aligns with crest and trough aligns with trough), they add up and create a larger wave with higher amplitude. This is called constructive interference. In contrast, when two waves meet out of phase (crest aligns with trough), they cancel each other out and create a smaller wave with lower amplitude. This is called destructive interference.

In the case of the red dots, the waves created by the two stones interfere constructively along certain directions, resulting in wave crests that are equally spaced. These crests form concentric circles. However, when these waves interfere in other directions, they create destructive interference, resulting in wave troughs. The points where the crests and troughs intersect form an ellipse.

On the other hand, the blue dots in the figure lie on a hyperbola because the waves created by the two stones interfere destructively.

Unlike the constructive interference seen in the red dots, the waves from the two stones interfere in such a way that their crests and troughs cancel each other out completely along certain directions. This creates regions where no waves are present, known as the nodal lines. The points lying on the nodal lines form a hyperbola. Therefore, the blue dots in the figure are located along a hyperbola due to destructive interference between the waves generated by the two stones.