Chapter 11.3, Problem 50E

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.

1. (a) Explain why the red dots lie on an ellipse.
2. (b) Explain why the blue dots lie on a hyperbola.

To determine

To explain: The reason for the red dots lie on an ellipse

Answer to Problem 50E

Because of for any red dot the sum of the distance from the two-fixed point remain same.

Explanation of Solution

Definition used:

Geometric definition of ellipse:

“An ellipse is the set of all points in the plane, the sum of whose distance from two fixed points $F_1$ and $F_2$ is a constant. These two fixed points are called foci of the ellipse.”

Description:

Given that two stone are dropped simultaneously into a calm pool of water.

The crests of the resulting wave form equally spaced concentric circles.

The center of the circles is fixed, say $F_1$ and $F_2$.

For any red dot, the sum of the number of crests from $F_1$ and number of crest from $F_2$ is equal to 16. That is, for any red dot, the sum of number of crest from $F_1$ and number of crest from $F_2$ is remain same as 16.

Therefore, for any red dot the sum of the distance from the two-fixed point remain same. Hence, by geometric definition of the ellipse stated above, conclude that, the red dots are on an ellipse with foci as $F_1$ and $F_2$(Centers of the concentric circles.)

To determine

To explain: The reason for the blue dots lie on a hyperbola.

Answer to Problem 50E

Because of for any red dot the differences of the distance from the two-fixed point remain same.

Explanation of Solution

Definition used:

Geometric definition of hyperbola:

“A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed points $F_1$ and $F_2$ is a constant. These two fixed points $F_1$ and $F_2$ are called the foci of the hyperbola.”

Description:

Given that two stone are dropped simultaneously into a calm pool of water.

The crests of the resulting wave form equally spaced concentric circles.

The center of the circles is fixed, say $F_1$ and $F_2$.

For any blue dot, the difference of the number of crests from $F_1$ and number of crest from $F_2$ is equal to 8. That is, for any blue dot, the difference of number of crest from $F_1$ and number of crest from $F_2$ is remain same as 8.

Therefore, for any blue dot the difference of the distance from the two-fixed point remain same. Hence, by geometric definition of the hyperbola stated above, conclude that, the red dots are on an ellipse with foci as $F_1$ and $F_2$(Centers of the concentric circles.).