How to Determine the Temperature Inside the Shower Room

Sometimes I sing in the shower. I can sometimes hear the echo when I hit particular notes.

The lowest note that I can sing, while simultaneously able to hear the reverbations is an A, 220Hz. We can model the shower room as an open-closed tube, in which resonant frequencies can be determined from the equation $f = \frac{nv}{4L}$, where $n$ is an odd number, $v$ is the speed of sound, and $L$ is length of tube. We know that the temperature inside the shower room is slightly higher than 25°C, so the speed of sound is expected to be slightly higher than 343m/s.

We can rearrange the equation to get $v = \frac{4fL}{n}$. Knowing that the glass panels that make up the shower room walls are 78” tall, we can convert that back to metric and say L = 1.98m. Plugging in n = 5 will obtain us a value of 348.5m/s.

From the textbook University Physics I found on OpenStax, equation 17.3.7 states that the speed of sound of air at sea level is about $v = 331 \sqrt{\frac{T}{273}}$, where T is temperature in Kelvins. Rearranging the equation gives us $T = 273 (\frac{v}{331})^2$, which evaluates to 303K = 30°C.

Now I know the ambient temperature of the shower room without using a thermometer, even though bringing in a thermometer is clearly a lot quicker.