Question

The resonance

frequency ( f ) of a closed air column is known to depend on the pressure ( p ), density ( rho ) of air, and the length of the air column ( l ). How are these quantities related?

a) ( f propto frac{p}{rho l} )
b) ( f propto frac{rho}{pl} )
c) ( f propto frac{pl}{rho} )
d) ( f propto frac{p l}{rho} )

Answer 1

The resonance frequency of a closed air column is influenced by the speed of sound, which is dependent on the temperature, and inversely proportional to the air column's length.

Step-by-step explanation:

The resonance frequency (f) of a closed air column depends upon several physical constants. Specifically, it is directly proportional to the speed of sound in the medium and inversely proportional to the length of the air column (L). In general, for a tube closed at one end, the relationship is given by f = nv / (4L), where n is an odd integer (1, 3, 5, ...), and v is the speed of sound in the medium, which is itself dependent on temperature, represented by the formula v = √(γ * (p/ρ)), with γ being the adiabatic index, p the pressure, and ρ the density of the air.

For instance, the fundamental frequency is observed when n=1, and the resonance occurs at a quarter of the wavelength. Therefore, the air temperature can affect the resonance frequency, as it influences the speed of sound.