The Speed of Sound — Question 2

Question 2

A temperature ratio must use an absolute scale

For a fixed ideal gas, use $c\propto\sqrt T$, with $T$ in kelvins. At $T_1=300\,\mathrm K$, $c_1=350\,\mathrm{m/s}$. The gas is heated uniformly to $T_2=363\,\mathrm K$. A source keeps frequency $700\,\mathrm{Hz}$.

1. Find the new speed and wavelength.

2. Find the temperature needed for a twenty-percent increase in speed from the initial state. Why is a twenty-percent temperature increase insufficient?

Question 2 - Solution

A temperature ratio must use an absolute scale

(a) Method and calculation

Taking the ratio removes the gas-dependent constant:$$c_2=c_1\sqrt{T_2/T_1}=350\sqrt{363/300}=385\,\mathrm{m/s}.$$$$\lambda_1=350/700=0.500\,\mathrm m,\qquad\lambda_2=0.550\,\mathrm m.$$

(b) Interpretation and further calculation

Inverting the square-root law gives$$\frac{T_2}{T_1}=\left(\frac{c_2}{c_1}\right)^2=1.20^2=1.44,\qquad T_2=432\,\mathrm K.$$A twenty-percent rise in absolute temperature gives only $\sqrt{1.20}-1=9.54\%$ more speed. Celsius temperature ratios are invalid because the model uses absolute temperature.

Physical check

Frequency is fixed by the driver; wavelength grows in the same ratio as speed.