Question

An oscillator vibrating at 1300 Hz produces a sound wave that travels through an ideal gas at 325 m/s when the gas temperature is 22.0 ∘C. For a certain experiment, you need to have the same oscillator produce sound of wavelength 28.9 cm in this gas.

What should the gas temperature be to achieve this wavelength?

Answer 1

In a given ideal gas the sound velocity a is a = const*sqrt(T) with T as absolute temperature in °K.

So T2 = T1*(a1/a2)²
On the other hand we have a = l*f

where l as wavelength and f frequency.

So l1*f1 = a1 and l2*f2 = a2
Given are: f1 = f2 = 1300 s⁻¹, a1 = 325 m/s, T1 = 273 + 22 =295 °K, l2 = 28.9 cm = 0.289 m
First we calculate l1 = a1/f = 325/1300 =0.25 m = 25 cm and a2 = l2*f2 = 0.289*1300 = 375.7 m/s.
Now T2 = T1*(a2/a1)² = 295*(375.7/325)² = 394.22 K = 121.22 ⁰C