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 will use the formula:
Velocity = Wavelength x Frequency
(a) How deep is the water at that location?
To find the depth of the water, we need to determine the time it took for the sound to travel to the ocean floor and back. Since the sound took 0.65 seconds to return, the round trip time is twice that amount. Therefore, the round trip time (T) is 0.65 s x 2 = 1.3 s.
We know that the speed of sound in water is 1500 m/s. We can use the equation to determine the depth (d):
Velocity = Speed of sound in water = 1500 m/s
Time = Round trip time = 1.3 s
d = (Velocity x Time) / 2
= (1500 m/s x 1.3 s) / 2
= 1950 m
Therefore, the depth of the water at that location is 1950 meters.
(b) What is the wavelength of the wave in water?
The formula for wavelength is:
Wavelength = Velocity / Frequency
Since the velocity of sound in water is 1500 m/s and the frequency emitted by the ship is 37 kHz, we need to convert the frequency to Hz:
Frequency = 37 kHz x 1000 = 37,000 Hz
Now, we can calculate the wavelength:
Wavelength = 1500 m/s / 37,000 Hz
To simplify the units, we can convert m/s to cm/s by multiplying by 100:
Wavelength = (1500 m/s x 100 cm/m) / 37,000 Hz
= 4054.05 cm / 37,000 s^(-1)
= 0.1096 cm
Therefore, the wavelength of the wave in water is approximately 0.1096 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?
To find the wavelength of the reflected wave in the air, we need to use the equation:
Wavelength = Velocity / Frequency
Since the velocity of sound in air is 350 m/s and the frequency emitted by the ship is still 37 kHz (after reflection), we convert the frequency to Hz as before:
Frequency = 37 kHz x 1000 = 37,000 Hz
Now, we can calculate the wavelength:
Wavelength = 350 m/s / 37,000 Hz
To simplify the units, we can convert m/s to mm/s by multiplying by 1000:
Wavelength = (350 m/s x 1000 mm/m) / 37,000 Hz
= 9.46 mm / 37,000 s^(-1)
= 0.0002554 mm
Therefore, the wavelength of the reflected wave as it travels into the air is approximately 0.0002554 mm.
To answer these questions, we need to use the equation for the speed of sound: speed = wavelength * frequency.
(a) To find the depth of the water, we'll use the fact that the time it takes for the sound wave to travel to the ocean floor and back is 0.65 s. Since the sound wave travels to the ocean floor and back, the distance it travels is twice the depth of the water. Using the equation for speed, we have:
speed = wavelength * frequency
Since the frequency is given as 37 kHz, we need to convert it to Hz by multiplying by 1000:
frequency = 37 kHz * 1000 = 37,000 Hz
The speed of sound in water is approximately 1500 m/s, so we can rewrite the equation as:
1500 m/s = wavelength * 37,000 Hz
To solve for wavelength, we can rearrange the equation:
wavelength = (1500 m/s) / (37,000 Hz)
Now we can substitute this value into the equation for distance:
distance = wavelength * 2
Substituting the value for wavelength gives us:
distance = ((1500 m/s) / (37,000 Hz)) * 2
Calculating this expression will give us the depth of the water in meters.
(b) To find the wavelength of the wave in water, we can use the equation:
wavelength = speed / frequency
Using the speed of sound in water (1500 m/s) and the frequency (37,000 Hz), we can calculate the wavelength in meters.
(c) To find the wavelength of the reflected wave as it travels into the air, where the speed of sound is 350 m/s, we can use the same equation as in part (b):
wavelength = speed / frequency
Using the speed of sound in air (350 m/s) and the frequency (37,000 Hz), we can calculate the wavelength in meters. Please note that we need to convert the speed of sound in air to the same unit as the speed of sound in water (m/s) before substituting it into the equation. When we have the wavelength in meters, we can convert it to millimeters by multiplying it by 1000.