Concept explainers
Repeal Prob. 7-112, but with the distance r from the sound source as an additional independent parameter.
(a)
A dimensionless relationship for I as a function of the other parameters by using the method of repeating variables in mass-based primary dimensions by using distance r from the sound source as an additional independent parameter.
Answer to Problem 113P
Dimensionless relationship between sound intensity and remaining parameter.
Explanation of Solution
Given Information:
Concept used:
The mass-based primary dimension will be used in this question. In this system all the possible variables are replaced by the mass. Mass, time length dimensions are represented as, [M],[L] and [t].
Concept of Buckingham's Pi method will also be used. It is represented as-
Where,
n= number of physical variables
k =independent physical quantities
n = total number of variable parameters
Calculation:
Primary dimensions of each parameter,
The total mass of parameters, n = 5
No. Of primary dimensions, j = 3
Expected no of
Dependant
Rewriting,
Mass,
Time,
Length,
Putting value in
Dependant Pi using independent variable P.
Mass,
Time,
Length,
Putting values in
Thus, from the equation of
Conclusion:
In this way, we are able to produce a dimensionless relationship for sound intensity using an independent parameter.
(b)
The expression for dimensionless relationship of I by using the force-based system of repeating variables by using distance r from the sound source as an additional independent parameter.
Answer to Problem 113P
Intensity by using force-based primary dimension system
For three repeating variables,
Explanation of Solution
Given:
Concept Used:
The force-based primary dimension will be used in this question. In this system all the possible variables are replaced by the mass. Force, time, length dimensions are represented as, [F], [L] and [t].
Concept of Buckingham's Pi method will also be used. It is represented as,
Where,
n= number of physical variables
k =independent physical quantities
n = total number of variable parameters
Calculation:
Now, the primary dimensions of all parameters are given below:
speed of sound =
density =
pressure level =
Sound Intensity,
The total mass of parameters, n = 5
No. Of primary dimensions, j = 3
Expected no of
Dependent p is calculated by using the I dependent variable.
Therefore,
By using primary dimensions,
For,
From equ(1) and equ (2),
Equating the exponents of both sides,
For force:
For time:
Putting the values of a1 and b1 in equation (1),
Dependent p is calculated by using P independent variable.
Therefore,
By using primary dimensions,
For,
From equ(3) and equ (4),
Equating the exponents of both sides,
For mass:
For time:
Putting the values of a2 and b2 in equation (3),
From equation (a) and (b),
Conclusion:
Intensity by using force-based primary dimension system.
For three repeating variables,
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Chapter 7 Solutions
Fluid Mechanics: Fundamentals and Applications
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