A ship mapping the depth of the ocean emits a sound of 37 kHz. The sound travels to the ocean floor and returns 0.65 s later.
(a) How deep is the water at that location?
m
(b) What is the wavelength of the wave in water?
cm
(c) What is the wavelength of the reflected wave as it travels into the air, where the speed of sound is 350 m/s?
mm
To answer these questions, we need to understand the relationship between the speed of sound, wavelength, and frequency.
The speed of sound in a medium can be calculated using the formula:
speed = wavelength × frequency
Let's start with the given information:
Frequency of the sound wave, f = 37 kHz = 37,000 Hz
Time taken for the sound wave to return, t = 0.65 s
(a) How deep is the water at that location?
To find the depth of the water, we need to calculate the distance traveled by the sound wave. Since the sound wave travels from the ship to the ocean floor and back, the total distance traveled is twice the depth of the water.
Distance = 2 × Depth
We can find the distance using the formula:
Distance = Speed × Time
Since the speed of sound in water is not given directly, we need to find it using the wavelength and frequency.
Step 1: Calculate the speed of sound in water.
speed = wavelength × frequency
Rearranging the formula, we get:
wavelength = speed / frequency
The speed of sound in water is approximately 1470 m/s.
Step 2: Calculate the distance.
Distance = Speed × Time
Distance = 1470 m/s × 0.65 s
Distance = 955.5 m
Now, calculating the depth of the water:
Depth = Distance / 2
Depth = 955.5 m / 2
Depth = 477.75 m
Therefore, the depth of water at that location is approximately 477.75 meters.
(b) What is the wavelength of the wave in water?
Using the formula obtained earlier:
wavelength = speed / frequency
wavelength = 1470 m/s / 37,000 Hz
wavelength ≈ 0.0398 m
To convert this wavelength to centimeters, multiply by 100:
wavelength = 0.0398 m × 100
wavelength ≈ 3.98 cm
Therefore, the wavelength of the wave in water is approximately 3.98 cm.
(c) What is the wavelength of the reflected wave as it travels into the air, where the speed of sound is 350 m/s?
The frequency of the wave remains the same as it travels from water to air, so we can use the same frequency value: 37,000 Hz.
Using the speed of sound in air (350 m/s) and the frequency, we can calculate the wavelength as:
wavelength = speed / frequency
wavelength = 350 m/s / 37,000 Hz
To convert this wavelength to millimeters, multiply by 1000:
wavelength ≈ 0.0095 m × 1000 = 9.5 mm
Therefore, the wavelength of the reflected wave as it travels into the air is approximately 9.5 mm.