On a distant planet, an alien creature sings a sustained musical note with a wavelength of 31.9 cm. Another alien produces a note with wavelength of 32.4 cm. The atmosphere near the planet's surface has a bulk modulus of 148 kN m-2 and its density is 1.37 kg m-3.
What beat frequency is heard when both aliens sing simultaneously?
To determine the beat frequency heard when both aliens sing simultaneously, we need to find the difference in frequencies between the two musical notes.
First, let's find the frequency of each note using the formula:
Frequency = Speed of Sound / Wavelength
The speed of sound in a medium is given by the formula:
Speed of Sound = √(Bulk modulus / Density)
Let's calculate the speed of sound using the given values:
Speed of Sound = √(148 kN m-2 / 1.37 kg m-3)
To simplify the calculation, we need to convert the bulk modulus from kN m-2 to N m-2:
Speed of Sound = √(148,000 N m-2 / 1.37 kg m-3)
Now, we can calculate the speed of sound.
Speed of Sound = √(108029.2 m2 s-2) ≈ 328.6 m/s
Now, let's calculate the frequency of each note:
Frequency1 = Speed of Sound / Wavelength1
Frequency1 = 328.6 m/s / 0.319 m = 1030 Hz
Frequency2 = Speed of Sound / Wavelength2
Frequency2 = 328.6 m/s / 0.324 m = 1013 Hz
To find the beat frequency, we subtract the lower frequency from the higher frequency:
Beat Frequency = Frequency2 - Frequency1
Beat Frequency = 1013 Hz - 1030 Hz ≈ -17 Hz
The negative sign indicates that the beat frequency is negative, which means the aliens are singing slightly out of phase, resulting in a beat frequency of approximately 17 Hz.
So, the beat frequency heard when both aliens sing simultaneously is approximately 17 Hz.
To find the beat frequency heard when both aliens sing simultaneously, you need to calculate the difference in frequencies between the two notes.
The formula to calculate the beat frequency is:
Beat frequency = |f1 - f2|
Where f1 and f2 are the frequencies of the two musical notes. Since the given information provides wavelengths, we first need to convert them into frequencies.
The formula to calculate frequency is:
Frequency = speed / wavelength
Where speed is the speed of sound in the given medium.
We can use the formula v = √(Bulk modulus / Density) to find the speed of sound in the planet's atmosphere.
Let's calculate the frequencies for the two notes:
For the first note:
Wavelength 1 = 31.9 cm
Speed of sound = √(Bulk modulus / Density) = √(148 kN/m^2 / 1.37 kg/m^3)
≈ √(108029.2 m^2/s^2)
≈ 328.45 m/s
Frequency 1 = speed / wavelength 1 = 328.45 m/s / 0.319 m = 1028.84 Hz
For the second note:
Wavelength 2 = 32.4 cm
Frequency 2 = speed / wavelength 2 = 328.45 m/s / 0.324 m = 1013.77 Hz
Now, let's calculate the beat frequency:
Beat frequency = |f1 - f2| = |1028.84 Hz - 1013.77 Hz| = 15.07 Hz
Therefore, the beat frequency heard when both aliens sing simultaneously is approximately 15.07 Hz.