Sound Intensity Formula

Sound intensity is also called acoustic intensity. The sound intensity formula equals the power carried by sound waves per unit area in a direction perpendicular to the path of sound waves.

Since the sound intensity equation carries two parameters, viz: power and the area, so the unit of sound intensity, i.e., ‘I’ is Watt per metre square. This unit can be written as Wm^-2. 

Another definition of the sound intensity level formula that we use for the decibel calculator intensity is as follows:

The formula of intensity of sound helps us calculate the noise sound intensity in the air at the listener’s position as a quantity of sound energy.

On this page, we will understand the sound intensity level formula, pressure amplitude formula, loudness of sound formula, and loudness and intensity formula.

Sound Intensity Level Formula

The formula of intensity of sound is:

                  \[I=\frac{P}{A}\]……….(1)

Here, 

P is the sound power in Watts, and

A is the area in m²

Besides the sound intensity formula, we have loudness and intensity formula, which is as follows:

                      \[I_{L}=10 log_{10}\frac{I}{I_{0}}\]

 Here,

I0  is the reference intensity A reference intensity has a value of 10^-12 Wm^-2. 

Also, we include the threshold of hearing between 1 kHz and 0 dB when calculating sound intensity levels in decibels.

(Image will be uploaded soon)

Fig: The above graph demonstrates sound pressures in two sound waves of different intensities. The source bearing large amplitude oscillations possess high sound intensity, also, has greater pressure maxima and minima. This happens because pressures are higher in the high-intensity sound, also, high pressure exerts larger forces on the objects it encounters.

Point To Note:

Though sound intensity is calculated in Wm^-2, this unit is much more referred to as decibels, dB, so that’s why we use the decibel calculator intensity in the loudness of sound formula.

Pressure Amplitude Formula

The pressure amplitude formula is:

                                \[I=\frac{\Delta P^{2}}{2\rho\upsilon (W)}\]……….(2)

Here,

P   =  Change in sound pressure or amplitude, calculated in Pascals  (Pa)

  =  Density of the material through which sound travels

vw   =   A speed of the observed sound

The above sound intensity equation (2) suggests that the larger your sound wave oscillation is, the more intense your sound will be.

Anther intensity of sound formula is as follows:

\[I=2\pi ^{2}\rho cv^{2}\delta ^{2}\]

Here,

v   = frequency of the sound

  =  amplitude. Amplitude is the maximum absolute value of sound wave (crest or trough)

c  = speed of sound in m/s

Please note that the definition of remains the same, as stated in the above equation (2).

Decibel Calculator Density

Decibel is a common measure of the intensity level of sound. It is one-tenth of bel on the logarithmic intensity scale, which is given as:

                  \[d_{B}=10 log_{10}\frac{P_{1}}{P_{2}}\]

This equation is also known as the intensity level of the sound formula.

Here,

P₁ and P₂ are the relative powers of sound.  

For understanding the sound intensity formula distance, we have the following intensity level of sound formula:

Sound Intensity Formula Distance

The loudness and intensity formula can be understood by the following intensity and the inverse square law:

The intensity and the inverse square law statement:

The inverse square law for sound applies to small point sources of sound that produce sound uniformly in all directions. Also, the medium through which sound travels is uniform such that sound neither reflects nor absorbed by or any other surface or the air.

Under all conditions, sound waves are spherical and they will spread evenly in all directions and their wavefronts will form a spherical surface.

Therefore, the power of the sound also spreads throughout this spherical surface. While finding the intensity of sound at a distance ‘r’ metres from the source, and the surface area of a sphere is \[\pi 4r^2\].

Since we know that the sound intensity formula is sound power divided by per unit area, so we put the value of the surface area in equation (1) as:

\[I=\frac{P}{4\pi r^{2}}\]…..(3)

Transposing equation (3), we can calculate the power of the source, which is:

\[P=I. 4\pi r^{2}\]

Now, referring back to eq (3), we see that sound intensity varies inversely with the square of the distance from the source, i.e.,

                       \[I\alpha \frac{1}{r^{2}}\]…..(4)

Now, let’s solve few sound intensity formula examples to understand its application:

Sound Intensity Formula Examples

Example 1: Megha measures the sound intensity at a distance of 6 m from the top of the tower to be 3.8 x 10^-2 Wm^-2. Let’s assume that the tip of the tower is a point source of the sound, ignoring the effects of reflection and absorption, calculate the total power of the sound emitted by the tower.

Given data:

r    =   6 m 

I   =  3.8 x 10^-2 Wm^-2

Solution:

From the equation: \[P=I. 4\pi r^{2}\]

Putting the value of r and I, we get:

                    =   (3.8 x 10^-2)* 4* 3.14 * (6)²

Solving the equation, we get the value of P as:

                             P    =   17.18 W

Answer: The tip of the tower produces a sound power of 17 W.

Example 2: If the intensity of the sound 4 m is 4.1 x 10^-2 Wm^-2 from the sound source, then find the intensity at 2.5 m.

Ans:

Given data:

I1    =  4.1 x 10^-2 Wm^-2

r1   =  4 m

r2   =  2.5 m

To find: I₂

From the eq: \[I\alpha \frac{1}{r^{2}}\]

Using the following relationship:

                     \[I_{1}r_{1}^{2}=I_{2}r_{2}^{2}\]

                      \[I_{2}=\frac{\ I_{1}r_{1}^{2}}{r^{2}}]

Putting the values in the above equation:

                  \[\frac{(4.1\times 10^{-2})*(4)^{2}}{2.5^{2}}\]

On solving, we get the value of  I2 as;

                     I₂    =  10.496  x 10^-2 Wm^-2

So, the intensity of the sound at a distance of 2.5 m is 10.496  x 10^-2 Wm^-2.

Overview of Sound Intensity and Decibels

For a reference point on the intensity of sound levels, below is the list of different intensities:

• 0 dB, I = 10^-12  -   This value corresponds to the threshold of human hearing

• 10 dB, I = 10^-11  -  This value corresponds to the sound produced by rustles of leaves.

• 60 dB, I = 10^-16  - This intensity of sound comes at the normal conservation.

• 100 dB, I = 10^-2  -  this sound intensity refers to the loud siren.

• 160 dB, I = 10⁴   -   The highest sound intensity can burst your eardrums.

Conclusion

The intensity of sound level formula helps us determine the intensity of sound at some distance ‘r’ metres from the source. Also, it helps us calculate the intensity level along with the sound power in Watts.

Although the unit of sound intensity is Watt per squared meter; the most preferred unit is decibels, abbreviated as dB. dB also tells the noise levels.