What is the frequency (in Hz) of the tone being emitted? (Assume air temperature to be 20°C.)

Question

A person stands directly in front of two speakers that are emitting the same pure tone. The person then moves to one side until no sound is heard. At that point, the person is 5.15 m from one of the speakers and 5.20 m from the other.

Answer 1

Hi Victoria!

Two signals will effectively cancel each other out if:

1) They are the same frequency

2) They are 180 degrees out of phase from each other (where one signal has a peak, the other has a trough and vice versa)

To be 180 degrees out of phase basically means the signal is completely inverted. At the specified location, these two signals will add up to form the wave that we hear. If they two waves are completely opposite of each other, they will cancel each other out.

It would be helpful to put two signals with the same frequency but different phase into a graphing calculator (such as desmos). Plot the following:

y = sin(2πx)

z = sin(2πx + π) note that π = 180 degrees

Lastly, add the two signals to create the end result

A = y + z

This should give you a flat line at 0

This problem basically asks: What wavelength of sound will be completely out of phase with itself after it travels an additional 0.05 m? I.e., what wavelength of a sound wave will be exactly one-half of a wavelength further along in its cycle when it travels the extra 0.05 meters? The answer is pretty straightforward: the full wavelength should be twice this difference in distance, so 0.1 meters.

For the last step, you just need to use the relationship of frequency f, wavelength λ, and speed of sound c:

f = c/λ

For c, use the speed of sound at 20 degrees C. It should be somewhere around 330 m/s, but look it up to be sure.

Hope this helps!