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 solve this problem, we can use the formula for the time it takes for sound to travel a certain distance:
Time = Distance / Speed
(a) To find the depth of the water, we need to calculate the distance traveled by the sound. The sound travels down to the ocean floor and then back up, so the total distance is twice the depth of the water.
The time taken for the sound to travel down and back up is 0.65 seconds.
We need to find the distance, so we rearrange the formula:
Distance = Time * Speed
Using the given speed of sound in water, which is approximately 1500 m/s:
Distance = 0.65 s * 1500 m/s = 975 m
Since the distance is twice the depth of the water, the depth of the water is half of 975 m:
Depth = 975 m / 2 = 487.5 m
So, the depth of the water at that location is 487.5 meters.
(b) To find the wavelength of the wave in water, we can use the equation:
Speed = Wavelength * Frequency
The speed of sound in water is approximately 1500 m/s, and the frequency emitted by the ship is 37 kHz, which is equivalent to 37,000 Hz.
Rearranging the formula, we have:
Wavelength = Speed / Frequency
Wavelength = 1500 m/s / 37,000 Hz
Wavelength ≈ 0.0405 meters or 4.05 cm
So, the wavelength of the wave in water is approximately 4.05 cm.
(c) To find the wavelength of the reflected wave as it travels into the air, we can use the same formula as in part (b):
Speed = Wavelength * Frequency
The speed of sound in air is given as 350 m/s, and the frequency emitted by the ship is still 37 kHz.
Rearranging the formula, we have:
Wavelength = Speed / Frequency
Wavelength = 350 m/s / 37,000 Hz
Wavelength ≈ 0.0095 meters or 9.5 mm
So, the wavelength of the reflected wave as it travels into the air is approximately 9.5 mm.
To find the answers to these questions, we need to use the formula for wave speed:
Wave speed = Frequency x Wavelength
Let's go through each question step by step:
(a) How deep is the water at that location?
To calculate the depth of the water, we need to use the formula:
Depth = (Speed of Sound x Time) / 2
Given that the speed of sound underwater is approximately 1500 m/s, and the time it takes for the sound to travel to the ocean floor and return is 0.65 s, we can substitute these values into the formula:
Depth = (1500 m/s x 0.65 s) / 2
Depth = 487.5 m / 2
Depth = 243.75 m
So, the water depth at that location is 243.75 meters.
(b) What is the wavelength of the wave in water?
To calculate the wavelength of the wave in water, we rearrange the formula for wave speed:
Wavelength = Wave speed / Frequency
Given that the frequency of the sound is 37 kHz (kilohertz), i.e., 37,000 Hz, and the speed of sound in water is approximately 1500 m/s, we can substitute these values into the formula:
Wavelength = 1500 m/s / 37000 Hz
Wavelength ≈ 0.04 meters
The wavelength of the wave in water is approximately 0.04 meters or 4 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?
When the sound wave travels from water to air, the speed of sound changes. To calculate the wavelength of the reflected wave, we use the same formula:
Wavelength = Wave speed / Frequency
Given that the speed of sound in air is approximately 350 m/s, and the frequency is still 37 kHz (37,000 Hz), we substitute these values into the formula:
Wavelength = 350 m/s / 37000 Hz
Wavelength ≈ 0.0095 meters
The wavelength of the reflected wave as it travels into the air is approximately 0.0095 meters or 9.5 mm.