Problem 46 - Algebra and Trigonometry

Question

Two stones are dropped simultaneously in a calm pool of water. The crests of the resulting waves form equally spaced concentric circles, as shown in the figures. The waves interact with each other to create certain interference patterns. (a) Explain why the red dots lie on an ellipse. (b) Explain why the blue dots lie on a hyperbola.

Step-by-Step Solution

Answer

Red dots form an ellipse due to a constant sum of distances from foci; blue dots form a hyperbola due to a constant difference in distances from foci.

Step 1: Understanding the Problem

When two stones are dropped simultaneously into the water, each stone creates a series of concentric circles (waves) that propagate outward. The interference patterns created by these overlapping waves help to determine where constructive and destructive interference occurs. Our task is to explain the patterns where these dots lie.

Step 2: Explaining the Ellipse for Red Dots

An ellipse is defined as the set of points where the sum of the distances from two fixed points (foci) is constant. Here, the two stones act as the foci. The red dots lie on paths where the sum of the distances from each red dot to both foci (points where stones were dropped) is the same. This consistent total path length forms an ellipse according to wave interference principles.

Step 3: Explaining the Hyperbola for Blue Dots

A hyperbola is defined as the set of points where the absolute difference in the distances from two fixed points (foci) is constant. For the blue dots, they lie on paths where the absolute difference in the distances from each blue dot to both foci is constant. This is characteristic of areas where destructive interference occurs, as waves subtract from one another to cancel the displacement at those dots.

Key Concepts

Ellipse

Think of an ellipse as a kind of squished circle. Imagine you have two pins in a board, and you tie a length of string between them. As you move a pencil around, keeping the string taut, you draw an ellipse. In this problem, the ellipse is formed by wave interference. The red dots all lie on an ellipse because each dot maintains a constant total distance from the two stones. These stones act like the pins in our string analogy.

The sum of the distances from each red dot to both stones is the same. This means if you measure the distance from the dot to one stone and add it to the distance from the dot to the other stone, it always equals some constant value. This is why the red dots trace out an elliptical path. Waves reach all these points in phase, causing constructive interference.

Hyperbola

A hyperbola might seem trickier than an ellipse at first, but it's quite fascinating. Imagine this: instead of using the sum, a hyperbola uses the difference between distances. Place two thumbtacks on your table again and think about measuring how different your distances to these from any point are.

The blue dots lie on a hyperbola because for each blue dot, the difference between its distances to the two stones is constant. These stones serve as the two critical points for defining a hyperbola, known as its foci. In our interference pattern, this means that the waves are meeting out of phase at these points, resulting in a cancellation, or destructive interference, making these the quiet or calm spots on the water.

Constructive Interference

Wave interference occurs when waves overlap, influencing each other's effects at certain points. Constructive interference is like waves working together; when they meet in phase, their effects add up! In the pond scenario with stones, the places where waves from both stones align perfectly to enhance each other are points of constructive interference. This is like two friends pushing a swing in the same direction at the same time—it goes higher!

At all points on the ellipse, the crest of a wave from one stone aligns with the crest of a wave from the other. This creates larger waves, leaving the red dots as markers of where this reinforcing pattern happens. The constructive interference results in visible peaks on the water surface, which is why the red dots appear on an ellipse in the interference pattern.

Destructive Interference

Destructive interference is where waves, instead of reinforcing each other's effects like in constructive interference, diminish each other. This happens when a wave crest meets a trough from another wave, leading them to cancel each other out. Think of it like trying to push a swing forward while a friend pushes backward—the swing stays pretty still.

In our water wave problem, destructive interference occurs along the hyperbola. At these blue dots, the crest of a wave from one stone perfectly matches up with a trough from the other stone. The cancellation means the water is much calmer there. The difference in distance to the stones leads to this out-of-phase overlap, which causes the waves to negate each other, marking the location of destructive interference perfectly along the hyperbolic paths.