How Deep is the Well When Humming Establishes Standing Waves?

[hbng02]

A person hums into the top of a well and finds that standing waves are established at frequency of 42,000, and 98 Hz. The frequency of 42 Hz. is not necessarily the fundamental frequency. The speed of sound is 343 m/s. How deep is the well?

 

[cookiemonster]

Originally posted by hbng02
... finds that stadning waves are established at frequency of 42,000, and 98 Hz.

What exactly do you mean by this?

cookiemonster

 

[M98Ranger]

To solve this problem, we can use the formula for the fundamental frequency of standing waves in a closed tube, which is f = nv/4L, where n is the harmonic number, v is the speed of sound, and L is the length of the tube. In this case, we are given the first and second harmonic frequencies (42,000 Hz and 98 Hz, respectively), so we can set up the following equations:

42,000 = (1)(343)/4L
98 = (2)(343)/4L

Solving for L in both equations, we get L = 0.002 m and L = 0.035 m, respectively. Since the length of the tube (or in this case, the depth of the well) cannot be negative, we can eliminate the first solution and conclude that the depth of the well is approximately 0.035 meters.