The ship in Figure P14.35 travels along a straight line parallel to the shore and a distance d = 500 m from it. The ship’s radio receives simultaneous signals of the same frequency from antennas A and B, separated by a distance L = 800 m. The signals interfere constructively at point C, which is equidistant from A and B. The signal goes through the first minimum at point D, which is directly outward from the shore from point B. Determine the wavelength of the radio waves.
To determine the wavelength of the radio waves, we can use the concept of interference.
Given:
Distance between the ship and shore (d) = 500 m
Distance between antennas A and B (L) = 800 m
Constructive interference occurs when the path difference between the waves from antennas A and B is an integer multiple of the wavelength (λ). The path difference between A and C is equal to the path difference between B and C.
From the given information, we can find the path difference between A and C by using the Pythagorean theorem:
Path difference (PAC) = √(d^2 + (L/2)^2)
Path difference between A and C = Path difference between B and C, so:
Path difference (PBC) = √(d^2 + (L/2)^2)
In order for the waves to interfere constructively at point C, the path difference must be an integer multiple of the wavelength:
Path difference (PAC) = n * λ
Path difference (PBC) = n * λ
Substituting the path differences, we have:
√(d^2 + (L/2)^2) = n * λ
√(d^2 + (L/2)^2) = n * λ
Squaring both equations to eliminate the square roots:
d^2 + (L/2)^2 = n^2 * λ^2
d^2 + (L/2)^2 = n^2 * λ^2
Subtracting the second equation from the first, we have:
0 = n^2 * λ^2 - n^2 * λ^2
Simplifying:
0 = n^2 * λ^2 - n^2 * λ^2
0 = 0
Since 0 = 0 is a true statement, it means that any value of λ will satisfy the equation. Therefore, the wavelength of the radio waves cannot be determined based on the given information.
To determine the wavelength of the radio waves, we can analyze the given information and apply the principles of wave interference. Here's how you can approach the problem:
1. Understand the scenario: In this scenario, the ship is receiving simultaneous signals from antennas A and B. These signals interfere constructively at point C and go through their first minimum at point D.
2. Identify the relevant points: Given the information, we can identify the following points:
- A: The position of antenna A
- B: The position of antenna B
- C: The point where the signals interfere constructively
- D: The point where the signals go through their first minimum
3. Analyze the interference pattern: Since the signals interfere constructively at point C, we can determine that the path lengths from A and B to point C are equal. Similarly, since the signal goes through its first minimum at point D, the path length from B to D is half a wavelength longer than the path length from A to D.
4. Use the given distances and variables:
- Distance between the ship and the shore: d = 500 m
- Distance between the antennas: L = 800 m
5. Apply the principles of wave interference:
- From point A to point C and point B to point C, the paths are equal. Therefore, the total path length from A to C and from B to C can be expressed as:
2d + L = λ/2, where λ is the wavelength
6. Solve for the wavelength:
- Rearrange the equation to solve for the wavelength:
λ = (2d + L) * 2
7. Substitute the given values:
- Substitute the values of d = 500 m and L = 800 m into the equation:
λ = (2*500 + 800) * 2
8. Calculate the wavelength:
- Perform the calculations:
λ = 1800 * 2
λ = 3600 m
9. State the final answer:
- The wavelength of the radio waves is 3600 meters.
By following these steps, you can determine the wavelength of the radio waves using the given information and principles of wave interference.