The air temperature is 25^circ C and an air column carries a standing sound wave at a frequency of 340 Hz . What is the length of the air column, which is closed at one end if you want to hear the third harmonic?K/U

Here's how to calculate the length of the air column:<br /><br />**Understanding the Concepts**<br /><br />* **Harmonics:** In a closed air column (closed at one end and open at the other), only odd harmonics are present. The first harmonic is the fundamental frequency, the third harmonic is three times the fundamental frequency, the fifth is five times, and so on.<br />* **Wavelength and Frequency:** The speed of sound (v) is related to its frequency (f) and wavelength (λ) by the equation v = fλ.<br />* **Wavelength in a Closed Air Column:** For a closed air column, the length (L) of the column is related to the wavelength of the standing wave by the equation L = (nλ)/4, where n represents the harmonic number (n = 1, 3, 5, ...).<br /><br />**Calculations**<br /><br />1. **Speed of Sound:** The speed of sound in air is temperature-dependent. A good approximation is given by v = 331.4 + 0.6T, where T is the temperature in Celsius. So, at 25°C:<br /><br /> v = 331.4 + 0.6 * 25 = 346.4 m/s<br /><br />2. **Wavelength of the Third Harmonic:** We are given that the frequency of the third harmonic (f₃) is 340 Hz. We can find the wavelength (λ₃) using the speed of sound (v) calculated above:<br /><br /> λ₃ = v / f₃ = 346.4 m/s / 340 Hz ≈ 1.019 m<br /><br />3. **Length of the Air Column:** For a closed air column and the third harmonic (n = 3), the length (L) is:<br /><br /> L = (3λ₃) / 4 = (3 * 1.019 m) / 4 ≈ 0.764 m<br /><br />**Answer:**<br /><br />The length of the air column needs to be approximately 0.764 meters to hear the third harmonic at 340 Hz and 25°C.<br />