Two speakers send out sound in phase at a frequency of 779 Hz. When you are the same distance from both speakers you hear a maximum sound. Moving around, you hear essentially no sound when you are 65 cm from one and 85 cm from the other, and again when you are 123 cm from one and 183 cm from the other. What is the speed of sound here?

Question

Two speakers send out sound in phase at a frequency of 779 Hz. When you are the same distance from both speakers you hear a maximum sound. Moving around, you hear essentially no sound when you are 65 cm from one and 85 cm from the other, and again when you are 123 cm from one and 183 cm from the other. What is the speed of sound here?

Answer 1

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Answer 2

To find the speed of sound in this scenario, we can use the concept of interference between the sound waves generated by the two speakers.

Let's call the distance between the two speakers "d" and the speed of sound "v". When you are at a point where you hear a maximum sound, it implies that the waves from both the speakers are in phase and constructively interfere. This occurs when the path difference between the two speakers is an integer multiple of the wavelength.

So, when you are equidistant from both speakers and hear a maximum sound, the path difference is:
Path difference = |distance from one speaker - distance from the other speaker|

In this case, the path difference is zero, as both distances are the same. Hence, the distance from one speaker equals the distance from the other speaker.

Now, let's consider the scenario when you are 65 cm from one speaker and 85 cm from the other speaker. At this distance, there is essentially no sound, which implies destructive interference. In this case, the path difference is equal to half of the wavelength.

So, Path difference = λ/2 = |distance from one speaker - distance from the other speaker|
= 85 cm - 65 cm
= 20 cm

Similarly, when you are 123 cm from one speaker and 183 cm from the other speaker, there is destructive interference, and the path difference is again equal to half of the wavelength.

Path difference = λ/2 = |distance from one speaker - distance from the other speaker|
= 183 cm - 123 cm
= 60 cm

To find the wavelength, we need to determine two distinct values of the path difference and use the relation:

λ = 2 * (distance with destructive interference)

Using the path differences we found:

λ = 2 * 20 cm = 40 cm
λ = 2 * 60 cm = 120 cm

Now, we can use the formula for speed:

v = f * λ

Given that the frequency is 779 Hz, we substitute the values:

v = 779 Hz * (40 cm / 100 cm)
v = 311.6 cm/s

Therefore, the speed of sound in this scenario is approximately 311.6 cm/s.