A tube of length 0.85 m has a fundamental frequency of 100 Hz when one end is closed. The speed of sound and resonance follow from the equation of motion.
Adjust conditions:
The sliders adjust the values
片端を閉じると基本振動数が100 Hzの管。両端を開くと何 Hzか？
両端を開いた管で、振動中も各部分の温度を一定に保つと基本振動数は？
The equations explain the phenomenon
1The pressure change in air is expressed in terms of the volume change.Δp=−K ΔVV……
2The equation of motion is applied to 1 layer.ρ ∂²u∂t²=−∂(Δp)∂x……
3The volume change is expressed in terms of displacement and substituted into the equation of motion.ρ ∂²u∂t²=K ∂²u∂x²……
4Comparison with the wave equation gives the speed of sound.v=√(Kρ)……
5K is calculated at constant temperature for air at 0 °C and 1 atmosphere.K=p⇒v=√(pρ)=280 m/s……
6K is calculated assuming no heat transfer.K=γ p⇒v=√(γ pρ)……
7Pressure and density are expressed in terms of temperature.v=√(γ R TM)……
8An approximate expression for the speed of sound is obtained in terms of the Celsius temperature t.First-order approximation:v≒331.3+0.61 t……
9The boundary conditions for air displacement are determined at both ends of the tube.u(0, t)=0，∂u∂x (L, t)=0……
10The resonance frequencies of a tube closed at one end are calculated.f=(2n − 1) v4L……
11Removing the cap gives a tube open at both ends, whose resonance frequencies are calculated.f=n v2L……
Further topics: beats, interference, and diffraction
12Sounds with slightly different frequencies are superposed.fbeat=|f₁ − f₂|……
13Sounds from 2 speakers are superposed.|r₁ − r₂|=(m + ½) λ……
14The spreading of waves passing through an opening is examined.sinθ≒λd……
What this chapter has shown
The speed of sound is v=√(K/ρ). For adiabatic changes in an ideal gas, K=γp, giving v=√(γRT/M). At the same temperature and composition, changing the pressure does not change the speed of sound.
200 HzThe fundamental frequency of a tube open at both ends
In the constant-temperature limit, the speed of sound is 287 m/s, and the fundamental frequency of the tube open at both ends is 169 Hz.
In everyday lifeBlowing across the mouth of a narrow-necked bottle produces sound mainly through Helmholtz resonance. Air in the neck moves as the air inside the bottle compresses and expands. Adding water reduces the volume of air inside the bottle and raises the pitch.
Detailed summary
The speed of sound depends on the air's stiffness and density. Its stiffness comes from the pressure produced by molecular collisions. During sound oscillations, there is no time for heat to escape, so the stiffness is γp. Overlooking this alone made Newton's prediction too low by roughly 2 tenths of the measured value; correcting it brings the prediction into close agreement with measurements. The pitch of the sound in a tube depends on the wavelengths allowed by the conditions at its ends and on the speed of sound. Beats, interference, and diffraction can be explained using the superposition principle introduced in the previous chapter.