About intensity of sound wave

[thuanthuan]

Suppose we have two source of disturbance to produce sound waves.
The intensity of sound produced by the first source at P is I1, and the other intensity is I2. So what is the intensity at point P.

I see many books saying that I=I1+I2. Can you explain why?

net medium displacement vector s=vector s1+vector s2
we have I=(1/2).p.v.(w.Sm)^2 p is the density of medium, v is sound speed, Sm is displacement amplitude of s

yet I1=(1/2).p.v.(w.S1m)^2, I2 =(1/2).p.v.(w.S2m)^2

obviously I doesn't equal to I1+I2 !

Thank you.

 

[LightHero]

Here is the explanation: The intensity of sound at point P is the sum of the intensities of the two sources, because the total displacement vector of the medium at point P is the sum of the displacements of each source. This is due to the superposition principle, which states that the total wave from several sources is the sum of all individual waves. Therefore, the intensity of the sound at P is the sum of the intensities of each individual source, i.e. I = I1 + I2.

 

[charng]

I can explain why the intensity of sound waves at point P is equal to the sum of the intensities produced by the two sources.

First, we need to understand that intensity is a measure of the energy carried by sound waves per unit area. It is directly proportional to the square of the amplitude of the sound wave. This means that if the amplitude of the sound wave is doubled, the intensity will increase by a factor of four.

Now, let's consider the two sources of disturbance producing sound waves. Each source will create its own sound wave with a certain amplitude. When these two waves meet at point P, they will interfere with each other. Depending on the phase relationship between the two waves, they can either reinforce or cancel each other out. This is known as interference.

If the two waves are in phase, meaning their peaks and troughs align, they will reinforce each other and the resulting sound wave at point P will have a higher amplitude. On the other hand, if the two waves are out of phase, they will cancel each other out and the resulting sound wave at point P will have a lower amplitude.

Now, let's go back to the equation for intensity: I = (1/2)ρvωS^2m, where ρ is the density of the medium, v is the speed of sound, ω is the angular frequency, and Sm is the amplitude of the sound wave.

Since intensity is directly proportional to the square of the amplitude, we can rewrite the equation as I ∝ S^2m. This means that the intensity at point P will depend on the total amplitude of the sound wave at that point, which is the sum of the amplitudes from the two sources.

Therefore, the equation I = I1 + I2 holds true because the total intensity at point P is the sum of the intensities produced by the two sources. It takes into account the interference between the two waves, resulting in a net intensity value.

In conclusion, the intensity at a point produced by two sources of disturbance is the sum of the individual intensities because it takes into account the interference between the two waves. This is why the equation I = I1 + I2 is valid.