Pressure equation of sound wave

[abhineetK]

Let us consider sound (longitudinal) wave
y=A.sin($$\omega$$t-kx)

See attachment for diagram

Consider air column of length $$\Delta$$x and cross-sectional area S at distance x from source.
I want to derive the pressure equation of the sound wave using
Bulk modulus, B=-$$\Delta$$P/($$\Delta$$V/V)
implying $$\Delta$$P=-B.$$\Delta$$V/V

But the trouble is that I am not able to select the expression for $$\Delta$$V
1) $$\Delta$$V=S.y
2) $$\Delta$$V=S$$\Delta$$y
TELL ME WHICH EXPRESSION OUT OF THE TWO SHOULD I USE with EXPLANATION.

 

[abhineetK]

I give you a hint about what I was actually thinking.

1)$$\Delta$$V=S.y
It would mean that $$\Delta$$P is the difference between the initial pressure of the volume considered and the final pressure of the same column (now volume is reduced to (V-$$\Delta$$V) but mass(amount) remains same).

2)$$\Delta$$V=S.$$\Delta$$y
It would mean that $$\Delta$$P is the change in pressure for a change $$\Delta$$y
in the length of the column.

Which of the above does the pressure equation of sound wave denotes? Please, tell.

NOTE: If you find any problem in understanding what I want to say, please tell.

 

[astrokreng]

The correct expression to use for \DeltaV is \DeltaV = S\Deltay, where \Deltay is the change in displacement of the air particles. This is because the change in volume of the air column is directly proportional to the change in displacement of the air particles. When a sound wave travels through a medium, it causes the air particles to vibrate back and forth in the direction of the wave's propagation. This vibration leads to a change in the volume of the air column, which can be represented by \DeltaV.

Using this expression for \DeltaV in the bulk modulus equation, we can derive the pressure equation of the sound wave as follows:

B = -\DeltaP/(\DeltaV/V) [Definition of bulk modulus]
B = -\DeltaP/(S\Deltay/V) [Substituting \DeltaV = S\Deltay]
B = -\DeltaP/(\Deltay/V) [S cancels out]
B = -V\DeltaP/\Deltay [Multiplying both sides by V]
V\DeltaP = B\Deltay [Rearranging terms]
\DeltaP = B\Deltay/V [Dividing both sides by V]

This is the pressure equation of the sound wave, which shows that the pressure change (\DeltaP) is directly proportional to the change in displacement of the air particles (\Deltay). It also shows that the bulk modulus (B) of the medium plays a role in determining the magnitude of the pressure change.

In summary, the expression \DeltaV = S\Deltay should be used because it accurately represents the relationship between the change in volume of the air column and the change in displacement of the air particles, in the context of a sound wave.