Figure 7.75 shows a tube structure in which a sound signal is bent from one end and is received at the other end . The semicircular part has a radius of `20.0 cm`. The frequency of the sound source can be varied electronically between `1000 and 4000 Hz`. Find the frequencies at which maxima of intensity are detected. The speed of sound in air ` = 340 m//s`.

Options

A. `1490 Hz and 2980 Hz`
B. `1490 Hz`
C. `2980 Hz`
D. none

Answer

A

Explanation

The sond wave reaches detector by two simultaneously by straight as well as semicircular track. The wave through the straight part travels a distance `l_(1) = 2 xx 20 cm` and the wave through the curved part travels a distance `l_(2) = pi (20 cm) = 62.8 cm` before they meet again and travel to the receiver . The path difference between the two waves received is , therefore
`Delta l = l_(2) - l_(1) = 62.8 cm - 40 cm = 22.8 cm = 0.228 m`
The wavelength of either wave is
`(v)/(n) = ( 340)/(n)`
For constructive interference , `Delta l = N//lambda`, where `N` is an integer.
`o.228 = N ((340)/(n))`
or `n = N ((340)/(0.228)) = N(1491.2) Hz = N (1490) Hz`.
Thus the frequencies within the specified range which cause maxima of intensity are `1490 Hz and 2980 Hz`.