Standing waves and maximum intensity?

[kathyt.25]

Homework Statement

"Two loudspeakers emit sound waves along the x-axis. The sound has maximum intensity when the speakers are 27.5 cm apart. The sound intensity decreases as the distance between the speakers is increased, reaching zero at a separation of 60.5 cm. What is the wavelength of the sound?"

I think this is more of a conceptual question that doesn't involve any actual calculation? I think it just involves an understanding of wavelengths and standing waves

Homework Equations

The Attempt at a Solution

I tried to work this one out, and this would be my reasoning, but it obviously doesn't fit in with the numbers given...
with my assumptions, I *should* be getting 1/2 wavelength = 27.5cm, and 1 wavelength = 55cm

So to work out this problem, I drew two sets of diagrams.
(1) The first diagram has speakers that are 27.5cm apart. Since this is when max intensity is generated, I figured that the antinode must be made in the middle, and that the waves shouldn't cancel out, and instead should be constructively superimposed - ie. added together without cancelling out. So in my diagram, the waves generated in this case make 1/2 a wavelength, which is 27.5cm.

(2) In the second diagram, Since this would be when intensity is zero, I thought that the two waves must cancel out... but still, this doesn't fit in with the assumption that the first case represents 1/2 wavelenght, and the second case represents a full wavelength

 

[earlofwessex]

I would guess that you need to consider the situation slightly differently.

assuming you had the speakers set up to produce a standing wave, then moved one of them further away by one full wavelength, what would happen? what would happen as you moved the speaker?

 

[nlightner]

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Overall, I'm not sure if I'm on the right track, but my understanding of standing waves is that they are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. The points where the waves are in phase (constructive interference) are called antinodes and the points where the waves are out of phase (destructive interference) are called nodes. In this scenario, the distance between the two loudspeakers is equal to the distance between two consecutive antinodes, which is equal to half a wavelength. This explains why the maximum intensity is observed when the speakers are 27.5 cm apart.

To find the wavelength, we can use the formula λ = 2d/n, where d is the distance between the speakers and n is the number of antinodes. In this case, n = 2 because there are 2 antinodes (one at each speaker) and d = 27.5 cm. Plugging in these values, we get λ = 2(27.5cm)/2 = 27.5 cm. This confirms our assumption that the first case represents half a wavelength.

As for the second case, when the speakers are 60.5 cm apart, the distance between two consecutive antinodes is equal to one wavelength. Therefore, the wavelength can be calculated as λ = 2d/n = 2(60.5cm)/1 = 60.5 cm. This also aligns with our understanding that when the speakers are 60.5 cm apart, the waves cancel out at the nodes, resulting in zero intensity.

In conclusion, the wavelength of the sound in this scenario is 27.5 cm. It is important to note that the wavelength of the sound remains constant, even as the distance between the speakers changes. This is because the frequency of the sound wave remains constant, and wavelength is inversely proportional to frequency. So, as the distance between the speakers increases, the number of antinodes decreases, but the wavelength remains the same.