Ocean waves pass through two small openings in a breakwater. This time the openings are d = 21 m apart. You're in a small boat on a perpendicular line midway between the two openings, L = 80 m from the breakwater. You row x = 32 m parallel to the breakwater and find yourself in a region of calm water. What is the wavelength of the ocean waves if the calm water you encounter at x = 32 m is the second calm region on your voyage from the center line?
Use the same equation as is used for the diffraction of light by a double slit. (Young's experiment)
http://theory.uwinnipeg.ca/physics/light/node9.html
To find the wavelength of the ocean waves, we need to use the concept of wave interference. When waves pass through two openings, they create regions of calm water where the waves interfere destructively.
In this scenario, the distance between the two openings is d = 21 m, and you are rowing parallel to the breakwater at a distance of x = 32 m from the center line. The calm water you encounter at x = 32 m is the second calm region on your voyage from the center line.
To determine the wavelength of the waves, we can use the concept of constructive and destructive interference. Constructive interference occurs when waves from the two openings interact and have the same phase, resulting in a higher amplitude. Destructive interference occurs when waves from the two openings interact and have opposite phases, resulting in a cancellation or reduction in amplitude.
In the case of the second calm region, we can assume that it is due to destructive interference. This means that the distance traveled by the waves from the two openings should be an integer multiple (n) of half the wavelength (λ/2) that creates destructive interference. Mathematically, we can express this as:
d + 2x = n(λ/2)
Substituting the given values:
21 m + 2 * 32 m = n(λ/2)
85 m = n(λ/2)
Here, we need to find the value of λ, which represents the wavelength. Therefore, we can rearrange the equation as:
λ = (2 * 85 m) / n
Since the calm region you encounter is the second one, you can assume n = 2. Substitute n = 2 into the equation:
λ = (2 * 85 m) / 2
λ = 85 m
Therefore, the wavelength of the ocean waves is 85 meters.
To find the wavelength of the ocean waves, we need to consider the wave interference pattern created by the two openings in the breakwater. Since the calm water region at x = 32 m is the second calm region on your voyage from the center line, it means that you are at a point where constructive interference occurs.
In constructive interference, the wave crests align, leading to regions of calm water. For this to happen, the path traveled by the waves from the two openings to your position must have a path difference of an integer number of wavelengths.
In this case, you are at a point L = 80 m from the breakwater, and you row x = 32 m parallel to the breakwater. The distance from the first opening to your position is (L - x / 2), and the distance from the second opening to your position is (L + x / 2). The path difference between the two waves is therefore (L + x / 2) - (L - x / 2) = x.
Since you encounter calm water at x = 32 m, it means that the path difference x corresponds to an integer number of wavelengths. Therefore, the wavelength of the ocean waves can be calculated by dividing the path difference x by the number of wavelengths.
In this case, we don't know the number of wavelengths, so let's assign it a variable n. The equation becomes:
x = n * wavelength
Rearranging the equation to solve for the wavelength:
wavelength = x / n
To find the value of n, we need to consider that you are at a point midway between the two openings. This means that the path difference should be equal to half of a wavelength (0.5 wavelength). Therefore:
x = 0.5 * wavelength
Solving for the wavelength:
wavelength = 2 * x
Plugging in the given value of x = 32 m, we can find the wavelength:
wavelength = 2 * 32 m = 64 m
Therefore, the wavelength of the ocean waves is 64 m.