Pressure Amplitude and Decreasing Intensity

[modulus]

When a point source emits sound, the sound travels away from the source as a series of wavefronts - all being spherical shells - away from the source right?

Now, we say the energy is conserved if we neglect damping forces in the medium, and so the power delivered by the source should be equal to the power delivered by a single wavefront at a certain distance from the source.

Power equals the intensity at a wavefront times the area of that wavefront. And, the intensity from a particular sound wave is proportional to the square of the pressure amplitude of the wave.

So, when a sound wave reaches a particular distance from the source, the pressure amplitude of the wave should decrease, as, at that distance, it is part of a wavefront, whose sound intesity is less than that of another wavefront closer to the source.

But, we also consider energy to be conserved for a single wave. But this is not possible, if we proceed by the above logic that explains why the pressure amplitude of a wave should decrease as it moves away from the source...?

This apparent paradox is going to make me mad...someone help, please!

 

[mathman]

You could write out some equations so we could see why you think energy is not conserved.

 

[Bobbywhy]

"To a good first approximation, wave energy is conserved as it propagates through the air. In a spherical pressure wave of radius , the energy of the wavefront is spread out over the spherical surface area . Therefore, the energy per unit area of an expanding spherical pressure wave decreases as . This is called spherical spreading loss. It is also an example of an inverse square law which is found repeatedly in the physics of conserved quantities in three-dimensional space. Since energy is proportional to amplitude squared, an inverse square law for energy translates to a decay law for amplitude."

https://ccrma.stanford.edu/~jos/pasp/Spherical_Waves_Point_Source.html

 

[modulus]

"To a good first approximation, wave energy is conserved as it propagates through the air. In a spherical pressure wave of radius , the energy of the wavefront is spread out over the spherical surface area . Therefore, the energy per unit area of an expanding spherical pressure wave decreases as . This is called spherical spreading loss. It is also an example of an inverse square law which is found repeatedly in the physics of conserved quantities in three-dimensional space. Since energy is proportional to amplitude squared, an inverse square law for energy translates to a decay law for amplitude."

Okay, so that explains exactly why amplitude should decrease as we move away from the source...it's according to the concept that the energy is spread out across the wavefront.

But, how does that translate in terms of linear sound waves? Pressure amplitude can't decrease as we move along a single longitudinal sound wave ,because energy is conserved along that single wave too.

?

 

[modulus]

Okay, about formulas:

The formula for the intensity (which is the power delivered by a wave per unit area) of a wave at any point on it (assuming all points along the wave have the same amplitude, frequency, etc.) is:

Intensity = 0.5 * (Density Of The Medium) * (Speed of Sound in the Medium) * (Angular Frequency of a Particle Oscillating on the Wave) * (Displacement Amplitude of a Particle Oscillating on the Wave)

For a single wave, none of these parameters changes along a wave.
But, considering the spherical wavefront explanation, the intensity decreases as we move farther away from the source (along a wave away from the source), and that demands that the amplitude decrease, as all the other parameters will remain constant for the single longitudinal wave - a point of which will constitute the wavefront at a certain distance from the source)...