an open tube is .86m long. Find the fundamental frequency and frequency of the next harmonic if the Temp=10deg celcius also if you closed one end of the tube what would the new fundamental frequency be?
I will be happy to critique your thinking.
To find the fundamental frequency and frequency of the next harmonic of an open tube, we can use the formula:
f = (n * v) / (2L)
Where:
- f is the frequency
- n is the harmonic number
- v is the speed of sound
- L is the length of the tube
First, let's determine the speed of sound at 10 degrees Celsius. The speed of sound in air can be calculated using the formula:
v = sqrt(gamma * R * T)
Where:
- v is the speed of sound
- gamma is the adiabatic index for air (approximately 1.4)
- R is the gas constant for air (approximately 287 J/(kg·K))
- T is the temperature in Kelvin
Converting 10 degrees Celsius to Kelvin:
T = 10 + 273.15 = 283.15 K
Now, let's calculate the speed of sound at 10 degrees Celsius:
v = sqrt(1.4 * 287 * 283.15) ≈ 340.29 m/s
For an open tube, the fundamental frequency corresponds to the first harmonic (n = 1). Plugging the values into the formula:
f1 = (1 * 340.29) / (2 * 0.86)
f1 ≈ 198.79 Hz
The frequency of the next harmonic (n = 2) can be calculated in the same way:
f2 = (2 * 340.29) / (2 * 0.86)
f2 ≈ 397.57 Hz
Now, if one end of the tube is closed, the fundamental frequency will change. For a closed tube, the fundamental frequency corresponds to the second harmonic (n = 2). The formula becomes:
f1_closed = (2 * v) / (4 * L)
f1_closed = (2 * 340.29) / (4 * 0.86)
f1_closed ≈ 197.95 Hz
In summary:
- Fundamental frequency (open tube): approximately 198.79 Hz
- Frequency of the next harmonic (open tube): approximately 397.57 Hz
- Fundamental frequency (closed tube): approximately 197.95 Hz