Question on the relationship between intensity of sound, amplitude and 1/r^2

[aznking1]

Homework Statement

Suppose sound wave is emitted uniformly in all directions by a speaker.

At a distance of 1.1m, the amplitude of it sound is 1.2x10^-8m

What is the amplitude of sound at a distance of 1.7m?

The Attempt at a Solution

What I'm confused is why does the amplitude of sound change? Isn't intensity = Power/Area? So at a greater distance, it just means that intensity is lowered. But power from a source is always constant isn't that right? And since power is proportional to amplitude squared, why does the amplitude decrease? am i missing out something?

The answer is 7.8x10^-9 using the relationship of intensity proportional to amplitude^2 and 1/r^2 if you are curious.

 

[mfb]

Isn't intensity = Power/Area? So at a greater distance, it just means that intensity is lowered.

Lower intensity at some specific point corresponds to a lower amplitude at that point. Meters as unit looks odd, I would expect Pa (pressure), but that should be irrelevant.

But power from a source is always constant isn't that right?

Constant in which way? I think you mean the total power here.

 

[aznking1]

Lower intensity at some specific point corresponds to a lower amplitude at that point. Meters as unit looks odd, I would expect Pa (pressure), but that should be irrelevant.

Constant in which way? I think you mean the total power here.

Ahh okay. I think i get it now after reading your post and thinking for so many hours.

Firstly, the formula intensity = power/area gives the power at a particular point. If we were to sum all the points ( surface area of a sphere), we would get the total power of the source, which is always constant

Hence, as distance increases, the intensity at a point decreases, meaning power at the specific point decreases, and so amplitude decreases, as you have mentioned.

Thank you! now i can get a good night's sleep haha... assuming i am correct

 

[mfb]

That is right.
Small issue: "Power at a specific point" is not meaningful. Intensity is. If you integrate that, you get a power.

 

[Nickie]

I can explain the relationship between intensity of sound, amplitude, and 1/r^2 to help clarify your confusion. First, let's define some terms:

- Intensity of sound: This refers to the amount of sound energy passing through a unit area per unit time. It is measured in watts per square meter (W/m^2).

- Amplitude: This is the maximum displacement of a wave from its equilibrium position. In the case of sound waves, it is the maximum displacement of air molecules from their resting position.

- Distance (r): This refers to the distance between the sound source and the point at which the sound is being measured.

Now, let's consider the relationship between these three variables. As you correctly mentioned, intensity is equal to power (the rate at which energy is transferred) divided by area. This means that as the distance from the sound source increases, the intensity decreases because the same amount of energy is spread out over a larger area. This is known as the inverse square law, which states that the intensity of a sound wave is inversely proportional to the square of the distance from the source (1/r^2).

Now, let's consider the relationship between intensity and amplitude. Intensity is directly proportional to the square of the amplitude. This means that as the amplitude of a sound wave increases, the intensity also increases. However, this relationship only holds true if the distance from the sound source remains constant. As we established earlier, as the distance from the sound source increases, the intensity decreases due to the inverse square law. Therefore, even if the amplitude of the sound wave remains constant, the intensity will decrease as the distance increases.

In summary, the amplitude of a sound wave does not change, but the intensity does change as the distance from the source increases. This is due to the inverse square law, which takes into account both the amplitude and distance from the source. So at a distance of 1.7m, the amplitude of the sound wave will be 7.8x10^-9m, which is lower than the amplitude at 1.1m due to the decrease in intensity caused by the increase in distance. I hope this helps clarify the relationship between intensity, amplitude, and 1/r^2.