Problem 50 - Algebra and Trigonometry

Question

Ripples in Pool Two stones are dropped simultaneously into 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 are on ellipses due to constant sum of distances to foci; blue dots are on hyperbolas due to constant difference.

Step 1: Understanding Wave Interference

When two wave sources (like stones dropped in water) create waves, the points where the waves have constructive interference (waves adding up) are on an ellipse. The distances from any point on an ellipse to the two foci (wave sources) add up to a constant.

Step 2: Condition for Red Dots on Ellipse

For the red dots, located on the crest intersections, the path difference between the waves from the two sources is such that it allows constructive interference. This path difference means that the sum of distances from any red dot to the two wave sources is constant, defining an ellipse.

Step 3: Defining a Hyperbola with Wave Paths

In contrast, blue dots lie on a hyperbola where the path difference is constant but corresponds to destructive interference. This means for the blue dots, the difference of distances from the wave sources to any blue dot remains constant.

Step 4: Condition for Blue Dots on Hyperbola

The wave paths leading to destructive interference create positions known as anti-nodal lines. For the blue dots, these are hyperbolic lines where the difference in distance from each source equals half a wavelength (destructive interference) or another odd multiple, defining a hyperbola.

Key Concepts

Constructive Interference

Constructive interference occurs when two waves meet and combine in a way that amplifies the resulting wave. This phenomenon happens when the crests (or troughs) of two waves align perfectly. In simpler terms, when waves overlap, they can add up to create a larger wave. This is visualized when you drop two stones into a pond, and the ripples that spread out from each stone interact with one another.

• When two waves have crests that coincide, they create a larger crest.
• Similarly, if their troughs coincide, they create a deeper trough.
• Their wave heights add together, resulting in a wave of increased amplitude.

In the context of the problem, the red dots lie at points of constructive interference, forming an ellipse. The consistent sum of distances to two wave sources leads to this constructive overlap, portraying a pattern of heightened wave activity. This pattern is visually apparent by the intensified points where their paths cross.

Destructive Interference

Destructive interference is the opposite of constructive interference. It occurs when two waves coincide in such a way that they cancel each other out. Instead of creating a larger wave, their energies subtract from one another. Imagine it as two people jumping on a trampoline on opposite sides, causing the trampoline to stay flat instead of bouncing.

• This happens when the crest of one wave aligns with the trough of another.
• Their amplitudes subtract from each other, leading to a smaller resultant wave or a flat line.
• In extreme cases, they can completely cancel out, resulting in no wave at all.

Regarding the exercise, the blue dots are where destructive interference occurs. This is because the path difference for these points creates conditions where the waves cancel each other out. The consistent difference in distances to the two sources forms a hyperbola, marking the locations of reduced or nullified wave energy.

Ellipse

An ellipse is a geometric figure that looks like a slightly squashed circle. It's defined by two points known as foci. A key property of an ellipse is that if you take any point on the ellipse, the total distance to these two foci is always constant.

• The sum of distances from any point on the ellipse to these foci is constant.
• This unique property makes ellipses essential in explaining wave phenomena.
• The red dots in the water exercise form an ellipse as the path lengths to the two wave sources are fixed and consistent.

The conceptual leap is realizing that such a simple geometric form can vividly illustrate complex interactions, such as wave interference. The way the waves constructively overlay at these distances creates the visible pattern of an ellipse.

Hyperbola

A hyperbola, unlike an ellipse, is a type of open curve with two branches. It's defined by its two foci and its characteristic difference in distances. At any point on a hyperbola, the difference in distance to its two foci is constant.

• Unlike ellipses, the difference in path length from any point on a hyperbola to its foci is consistent.
• Hyperbolas are open curves and extend indefinitely, unlike the closed loop of an ellipse.
• The blue dots from the wave exercise follow this hyperbolic path because the waves are destructively interfering, with consistent path differences.

In the water ripple exercise, this path represents areas of destructive interference. At these locations, the waves cancel each other out, leading to areas of lower energy or quiet spots. The hyperbola beautifully captures this dynamic balance of wave interactions, showcasing complex wave behavior in a form that's geometrically straightforward.