Pressure of Sound Wave: Inverse Distance Law

[vin300]

It is not difficult to imagine why the intensity of a sound wave would follow inverse square law, as the spherical area increases as square of radius, all point sources of gravity, electrostatics and many more sources of energy follow the same law. When you think of Newtonian gravitational force or electric force, all the formulae are classically parallel. What stands out, and therefore troubles me, is that the pressure of a wave, such as sound wave is said to follow a different proportionality, that of inverse distance law. That is the law followed by the energy terms because of integration, but they say that incase of sound, the intensity term follows inverse square while the pressure follows r^(-1) which is the whole thing in reverse connection. Being a man of science does come along with a man of doubt.

 

[nasu]

For any spherical wave the amplitude decreases as 1/r and intensity as 1/r^2.
Why do you think there is something special about pressure in a sound wave? Intensity is proportional to pressure squared.

 

[tech99]

For any spherical wave the amplitude decreases as 1/r and intensity as 1/r^2.
Why do you think there is something special about pressure in a sound wave? Intensity is proportional to pressure squared.

Can you qualitatively explain for a person with the brain of a five year old why amplitude follows 1/r? Many thanks.

 

[nasu]

I don't think I can and I don't think there is any point in trying. At that age you are interested in other things.:)

 

[tech99]

I don't think I can and I don't think there is any point in trying. At that age you are interested in other things.:)

Thank you very much for that. My problem is that I always think of the power density falling with the inverse square law, then derive the amplitude from that. But I cannot seem to explain to myself from first principles why the amplitude falls with 1/r. It is clearly important, because it distinguishes the radiation fields of an antenna from the induction fields.

 

[vin300]

Let's think this way. The energy is being carried across the spherical surface and has to be same across all spheres, so density prop to inverse square radius. The oscillator pressure or force is proportional to mean particle displacement, so across all spheres, the mean force and mean displacement must follow the same proportionality. The product yields 1/r^2, the two equal variables yielding a product 1/r^2 has to be 1/r and 1/r, so both pressure and particle displacement proportional to 1/r, whose product is energy flux.

 

[tech99]

Let's think this way. The energy is being carried across the spherical surface and has to be same across all spheres, so density prop to inverse square radius. The oscillator pressure or force is proportional to mean particle displacement, so across all spheres, the mean force and mean displacement must follow the same proportionality. The product yields 1/r^2, the two equal variables yielding a product 1/r^2 has to be 1/r and 1/r, so both pressure and particle displacement proportional to 1/r, whose product is energy flux.

I think this re-states the answer given by Nasu in post no. 2, which was suggesting that the intensity should be obtained by squaring the amplitude. But how does one obtain the 1/r amplitude law without starting from intensity? And further to that, at a simple level, and not deriving it from intensity, why does the electric field strength of an EM wave fall as 1/r whilst that of a charge falls as 1/r^2?