Relation between Octave bands and Overall noise level

RELATION BETWEEN OCTAVE BANDS AND OVERALL NOISE LEVEL

 

Introduction

The word ‘octave’ is derived from the Latin word meaning ‘eight’. In the musical world where there are 7 notes, the 8^th note sounds twice as high as the 1^st note. The 8^th note is an octave higher than the 1^st note. Similarly in the octave band, the upper limit frequency is twice the lower limit frequency. Octave bands are very useful in engineering applications because they reveal the spectral content, meaning they represent the change in noise levels with respect to the frequency of sound. This helps in identifying which frequency is responsible for the noise which helps in nailing down the component in a machine responsible for the particular frequency.

Overall noise level is a single number value that describes a noise source. The overall noise level is the average of all individual noise levels emitted by the noise source. Since it is a single number value, it does not provide any information on the frequency content of the noise. However it is very useful to characterize and rank noise sources based on the overall noise level.

The Octave Band

Consider a machine which emits a steady noise. The noise emitted by the machine is captured by a suitable Real Time Analyzer (RTA) for a period of 10 second. The acquired raw data is then processed and the result is displayed as an Octave 1/1 plot.

Equations

The Sound pressure level equation is given by,

Where,

X_i – Sound pressure level in dB(A)

P_i – Sound pressure in Pa

P₀ – Reference sound pressure (P₀ = 20 x 10^-6 Pa)

 

The logarithmic addition or the total sound pressure level is given by,

X₁, X₂… X_n – Individual Sound pressure levels in dB(A)

X – Total Sound pressure level in dB(A)

 

Calculation

The average sound pressure level (SPL) in each octave can be inferred from the Octave 1/1 band considered in this example. The total SPL or the overall noise level is determined by logarithmically adding the individual SPL values.

Octave band tabular column

Frequency band (Hz)	Sound pressure level (dB(A))
31.5	30
63	35
125	40
250	35
500	55
1000	70
2000	50
4000	42
8000	26

 

Plugging in the individual SPL values from above tabular column in equation 2,

This is the overall noise level of the considered noise source. Notice that the overall level is very close to the peak sound pressure level from the tabular column which is 70 dB(A).

Graph

The overall noise level is a plot of sound pressure level versus time. Since the noise source is steady with respect to time, the overall noise level is constant and does not fluctuate.

Conclusion

Thus by logarithmically adding the individual sound pressure levels, the overall noise level is obtained. The overall level is very useful as it quantifies a noise source in a single value. The overall level does not offer any insight into the frequency composition of the sound which is a drawback. Two noise sources can have the same overall level but can sound pleasant and harsh. This is because frequency content and human hearing play a major role in determining the sound quality. For example, a noise source with lot of high frequency content would sound unpleasant despite having the same overall level as a source with less high frequency content.

Projectile motion on Neutron star

PROJECTILE MOTION ON NEUTRON STAR

INTRODUCTION

Projectile motion is a form of motion that follows or traces out a parabolic path. The predominant reason for the origin of projectile motion is acceleration due to gravity. When an object is simply given a horizontal initial velocity, gravity which is always acting downward will exert a vertical pull on the object. The resultant of horizontal and vertical components is a parabolic motion. The magnitude of horizontal and vertical components may or may not be equal since it depends on the angle of trajectory. However both horizontal and vertical components are totally independent of each other. In this post, we intend to determine time of flight, maximum attainable height and maximum attainable distance of an object undergoing projectile motion on the surface of a Neutron star. 

ASSUMPTIONS

1. Air, wind and other frictional resistance are neglected

2. Effect of rotation of Neutron star is negligible

3. Temperature effects do not impede the motion

4. The ground surface is perfectly horizontal

5. The projectile moves along a two dimensional path

6. The projectile is indestructible

7. The motion is always parabolic without any effects from space-time distortion

CALCULATION

Consider an object of mass ‘M’ kg, moving with an initial velocity ‘u’ at an angle ‘θ’ with respect to the horizontal. Let ‘g’ be the acceleration due to gravity on Neutron star. It is important to note that projectile motion is independent of the mass of the object in a vacuum. However in air or other media, the drag coefficient being different for various object shapes and sizes, it is no longer independent of mass.

The equation of projectile motion in this case is given by,

where,

h – Horizontal distance at which the projectile attains maximum height (m)

k – Maximum height attained by the projectile (m)

a – Focal length of the parabola (m)

A projectile motion is represented in figure.1 with all the coordinates

Fig .1 A projectile motion

Now, the other aspects of the parabolic motion such as total time taken, maximum height and maximum distance attained will be discussed

The time of flight ‘T’ is given by,

where,

T – Time of flight or total time taken (s)

u – Initial velocity of projectile (m/s)

θ – Angle of projectile (degree)

g – Acceleration due to gravity on Neutron star (7 * 10¹² m/s²)

The maximum height attained (H) by the object is given by,

The maximum distance attained (d) by the object is given by,

From equations (3), (5) and (7) it can be observed that the values are extremely small. This is predominantly due to the fact that the Neutron star exerts extreme gravitational pull on the projectile. Although we assumed that the projectile is indestructible, it will complete the motion in an instant.

GRAPH

In order to plot the projectile motion, substitute equations (5) and (7) in equation (1) which is the equation of the projectile motion.

Since the curve passes through the origin, it must satisfy the origin or in other words the origin is a trivial solution of the above equation. Thereby substituting (x, y) as (0, 0) in the above equation, we can determine the value of the focal length which is a constant.

Now substitute equation (11) in equation (9) in order to plot the projectile motion.

Input the above equation in a suitable equation or curve plotter and the corresponding result will be obtained as shown in figure.2

Fig .2 Projectile motion on Neutron star

CONCLUSION

Thus the time of flight, maximum attainable height and distance of an object undergoing parabolic motion on the surface of a Neutron star were determined successfully. The plot also verifies a proper projectile motion with the maximum attained height and distance.

Projectile motion on Sun

PROJECTILE MOTION ON SUN

INTRODUCTION

Projectile motion is a form of motion that follows or traces out a parabolic path. The predominant reason for the origin of projectile motion is acceleration due to gravity. When an object is simply given a horizontal initial velocity, gravity which is always acting downward will exert a vertical pull on the object. The resultant of horizontal and vertical components is a parabolic motion. The magnitude of horizontal and vertical components may or may not be equal since it depends on the angle of trajectory. However both horizontal and vertical components are totally independent of each other. In this post, we intend to determine time of flight, maximum attainable height and maximum attainable distance of an object undergoing projectile motion on the surface of Sun.

ASSUMPTIONS

1. Air, wind and other frictional resistance are neglected

2. Effect of rotation of Sun is negligible

3. Temperature effects do not impede the motion

4. The ground surface is perfectly horizontal

5. The projectile moves along a two dimensional path

6. The projectile is indestructible

CALCULATION

Consider an object of mass ‘M’ kg, moving with an initial velocity ‘u’ at an angle ‘θ’ with respect to the horizontal. Let ‘g’ be the acceleration due to gravity on Sun. It is important to note that projectile motion is independent of the mass of the object in a vacuum. However in air or other media, the drag coefficient being different for various object shapes and sizes, it is no longer independent of mass.

The equation of projectile motion in this case is given by,

where,

h – Horizontal distance at which the projectile attains maximum height (m)

k – Maximum height attained by the projectile (m)

a – Focal length of the parabola (m)

A projectile motion is represented in figure.1 with all the coordinates

Fig .1 A projectile motion

Now, the other aspects of the parabolic motion such as total time taken, maximum height and maximum distance attained will be discussed

The time of flight ‘T’ is given by,

where,

T – Time of flight or total time taken (s)

u – Initial velocity of projectile (m/s)

θ – Angle of projectile (degree)

g – Acceleration due to gravity on Sun (273.7 m/s²)

The maximum height attained (H) by the object is given by,

The maximum distance attained (d) by the object is given by,

GRAPH

In order to plot the projectile motion, substitute equations (5) and (7) in equation (1) which is the equation of the projectile motion.

Since the curve passes through the origin, it must satisfy the origin or in other words the origin is a trivial solution of the above equation. Thereby substituting (x, y) as (0, 0) in the above equation, we can determine the value of the focal length which is a constant.

Now substitute equation (11) in equation (9) in order to plot the projectile motion.

Input the above equation in a suitable equation or curve plotter and the corresponding result will be obtained as shown in figure.2

Fig .2 Projectile motion on Sun

CONCLUSION

Thus the time of flight, maximum attainable height and distance of an object undergoing parabolic motion on the surface of Sun were determined successfully. The plot also verifies a proper projectile motion with the maximum attained height and distance.

Projectile motion on Jupiter

PROJECTILE MOTION ON JUPITER

INTRODUCTION

Projectile motion is a form of motion that follows or traces out a parabolic path. The predominant reason for the origin of projectile motion is acceleration due to gravity. When an object is simply given a horizontal initial velocity, gravity which is always acting downward will exert a vertical pull on the object. The resultant of horizontal and vertical components is a parabolic motion. The magnitude of horizontal and vertical components may or may not be equal since it depends on the angle of trajectory. However both horizontal and vertical components are totally independent of each other. In this post, we intend to determine time of flight, maximum attainable height and maximum attainable distance of an object undergoing projectile motion on the surface of Jupiter.

ASSUMPTIONS

1. Air, wind and other frictional resistance are neglected

2. Effect of rotation of Jupiter is negligible

3. Temperature effects do not impede the motion

4. The ground surface is perfectly horizontal

5. The projectile moves along a two dimensional path

6. The projectile is indestructible

CALCULATION

Consider an object of mass ‘M’ kg, moving with an initial velocity ‘u’ at an angle ‘θ’ with respect to the horizontal. Let ‘g’ be the acceleration due to gravity on Jupiter. It is important to note that projectile motion is independent of the mass of the object in a vacuum. However in air or other media, the drag coefficient being different for various object shapes and sizes, it is no longer independent of mass.

The equation of projectile motion in this case is given by,

where,

h – Horizontal distance at which the projectile attains maximum height (m)

k – Maximum height attained by the projectile (m)

a – Focal length of the parabola (m)

A projectile motion is represented in figure.1 with all the coordinates

Fig .1 A projectile motion

Now, the other aspects of the parabolic motion such as total time taken, maximum height and maximum distance attained will be discussed

The time of flight ‘T’ is given by,

where,

T – Time of flight or total time taken (s)

u – Initial velocity of projectile (m/s)

θ – Angle of projectile (degree)

g – Acceleration due to gravity on Jupiter (24.5 m/s²)

The maximum height attained (H) by the object is given by,

The maximum distance attained (d) by the object is given by,

GRAPH

In order to plot the projectile motion, substitute equations (3), (5) and (7) in equation (1) which is the equation of the projectile motion.

Since the curve passes through the origin, it must satisfy the origin or in other words the origin is a trivial solution of the above equation. Thereby substituting (x, y) as (0, 0) in the above equation, we can determine the value of the focal length which is a constant.

Now substitute equation (11) in equation (9) in order to plot the projectile motion.

Input the above equation in a suitable equation or curve plotter and the corresponding result will be obtained as shown in figure.2

 Fig .2 Projectile motion on Jupiter

CONCLUSION

Thus the time of flight, maximum attainable height and distance of an object undergoing parabolic motion on the surface of Jupiter were determined successfully. The plot also verifies a proper projectile motion with the maximum attained height and distance.

Projectile motion inside International Space Station (ISS)

PROJECTILE MOTION INSIDE INTERNATIONAL SPACE STATION

INTRODUCTION

Projectile motion is a form of motion that follows or traces out a parabolic path. The predominant reason for the origin of projectile motion is acceleration due to gravity. When an object is simply given a horizontal initial velocity, gravity which is always acting downward will exert a vertical pull on the object. The resultant of horizontal and vertical components is a parabolic motion. The magnitude of horizontal and vertical components may or may not be equal since it depends on the angle of trajectory. However both horizontal and vertical components are totally independent of each other. In this post, we intend to determine time of flight, maximum attainable height and maximum attainable distance of an object undergoing projectile motion inside the International Space Station (ISS).

ASSUMPTIONS

1. Air, wind and other frictional resistance are neglected

2. Temperature effects do not impede the motion

3. The ground surface is perfectly horizontal

4. The projectile moves along a two dimensional path

CALCULATION

Consider an object of mass ‘M’ kg, moving with an initial velocity ‘u’ at an angle ‘θ’ with respect to the horizontal. Let ‘g’ be the acceleration due to gravity on ISS. It is important to note that projectile motion is independent of the mass of the object in a vacuum. However in air or other media, the drag coefficient being different for various object shapes and sizes, it is no longer independent of mass. The ISS is always in a state of free fall, hence experiences weightlessness. Thereby all the objects inside the ISS will also experience weightlessness. Hence it is safe to assume that the acceleration due to gravity is zero.

The equation of projectile motion in this case is given by,

where,

h – Horizontal distance at which the projectile attains maximum height (m)

k – Maximum height attained by the projectile (m)

a – Focal length of the parabola (m)

A projectile motion is represented in figure.1 with all the coordinates

Fig .1 A projectile motion

Now, the other aspects of the parabolic motion such as total time taken, maximum height and maximum distance attained will be discussed

The time of flight ‘T’ is given by,

where,

T – Time of flight or total time taken (s)

u – Initial velocity of projectile (m/s)

θ – Angle of projectile (degree)

g – Perceived acceleration due to gravity inside ISS (0 m/s²)

The value ‘ꝏ’ or ‘infinity’ implies that the projectile motion will never be completed because due to lack of gravity, the projectile will continue to move at 45° rather than falling down to the surface.

The maximum height attained (H) by the object is given by,

The value ‘ꝏ’ indicates that the object will never achieve any maximum height since the projectile will always keep moving upward at the launch angle instead of falling down.

The maximum distance attained (d) by the object is given by,

The value ‘ꝏ’ indicates that the object will never achieve any maximum distance since the projectile will always keep moving upward at the launch angle instead of falling down.

Equations (3), (5) and (7) indicate that projectile motion cannot be completed inside the ISS since it is always in a state of free fall.

CONCLUSION

Thus it can be observed from the calculations that the projectile motion can never be completed inside the ISS.