The lower frequency limit for human hearing is usually considered to be 20.0 Hz. (a) What is
the speed of sound in m/s if the air temperature is 20.0ºC? (b) What is the corresponding
wavelength for this frequency?
(a) The speed of sound in air at a given temperature can be calculated using the following formula:
v = 331.4 + 0.6 * T
where v is the speed of sound in m/s, and T is the air temperature in ºC.
For an air temperature of 20.0ºC:
v = 331.4 + 0.6 * 20.0
v ≈ 343.4 m/s
The speed of sound in air at 20.0ºC is approximately 343.4 m/s.
(b) To find the wavelength corresponding to the lower frequency limit for human hearing (20.0 Hz), we can use the formula:
λ = v / f
where λ is the wavelength, v is the speed of sound in m/s, and f is the frequency in Hz.
λ = 343.4 m/s / 20.0 Hz
λ ≈ 17.17 m
The corresponding wavelength for the lower frequency limit for human hearing at 20.0ºC is approximately 17.17 meters.
To find the answers to these questions, we need to use the formulas related to the speed of sound and the wavelength of a wave.
(a) To calculate the speed of sound, we can use the formula:
Speed of sound = wavelength × frequency
Since we know the frequency (20.0 Hz), we need to find the wavelength at this frequency.
(b) The formula to calculate wavelength is:
Wavelength = speed of sound / frequency
To find the speed of sound, we can use the equation:
Speed of sound = 331.4 m/s + (0.6 m/s/°C) × (temperature in °C)
Given that the temperature is 20.0ºC, we can substitute it into the formula to find the speed of sound.
To summarize the steps to find the speed of sound and the corresponding wavelength:
Step 1: Calculate the speed of sound using the formula: Speed of sound = 331.4 m/s + (0.6 m/s/°C) × (temperature in °C)
Step 2: Calculate the wavelength using the formula: Wavelength = speed of sound / frequency
Now, let's calculate these values:
Step 1:
Speed of sound = 331.4 m/s + (0.6 m/s/°C) × (20.0ºC)
Speed of sound = 331.4 m/s + (0.6 m/s/°C) × (20.0)
Speed of sound = 331.4 m/s + 12 m/s
Speed of sound = 343.4 m/s
Therefore, the speed of sound is 343.4 m/s when the air temperature is 20.0ºC.
Step 2:
Wavelength = speed of sound / frequency
Wavelength = 343.4 m/s / 20.0 Hz
Wavelength = 17.17 meters (rounded to two decimal places)
Therefore, the corresponding wavelength for a frequency of 20.0 Hz and air temperature of 20.0ºC is approximately 17.17 meters.
To find the speed of sound in m/s at an air temperature of 20.0ºC and the corresponding wavelength for a frequency of 20.0 Hz, we can use the following formulas:
(a) The speed of sound in air can be calculated using the formula:
v = √(γ * R * T)
where:
v = speed of sound in m/s
γ = specific heat ratio of air (approximately 1.4)
R = gas constant (approximately 8.314 J/(mol·K))
T = air temperature in Kelvin
The given air temperature is 20.0ºC, which is equivalent to 293.15 K. Plugging in the values:
v = √(1.4 * 8.314 * 293.15)
Calculating:
v ≈ √(3344.628) ≈ 57.94 m/s
Therefore, the speed of sound in air at 20.0ºC is approximately 57.94 m/s.
(b) The corresponding wavelength for a frequency can be calculated using the formula:
λ = v / f
where:
λ = wavelength in meters
v = speed of sound in m/s
f = frequency in Hz
Plugging in the values:
λ = 57.94 / 20.0
Calculating:
λ ≈ 2.897 m
Therefore, the corresponding wavelength for a frequency of 20.0 Hz is approximately 2.897 meters.