Answer: E=ρV^3f(nr/V)
Step-by-step explanation:
The power per unit cross sectional area (E)
Transmitted by a sound wave is a function of of wave speed (V)
Medim density("ρ")
Wave amplitude (r)
Frequency (n)
E=f(V, ρ, r, n)
There are 5 parameters, that is, n=5
The primary variables M, L, T, m=3
Therefore, the number of repeating variables, r=3.
The repeating variables ρ, v, r.
According to Buckingham there will be n-m dimensionless group which are π1 and π2.
The dimensions for parameters are:
E= MT^-3
V= LT^-1
ρ= ML^-3
r=L
n= T^-1
π1= ρ^aV^br^cE
= (ML^-3) ^a(LT^-1) ^b(L) ^c
=M^a+1L^-3a+b+cT^-b-3
=M^0L^0T^0
By equating the coefficient
M: a+1= 0
a=-1
T: - b-3
b=-3
L: -3a+b+c= 0
3-3+c=0
c=0
π1=ρ^-1V^-3r^0E
π1= E/ρV^3
Check for dimensions
π1=MT^-3/(ML^-3)(LT^-3)
π1=1
π2=ρ^dV^er^e n
(ML^-3) ^d(LT^-1) ^e(L) ^f(T^-1)
=M^0T^0L^0
M: d=0
T:-e-1=0
e=-1
f:-3d+e+f=0
f=1
π2 =ρ^0V^-1r^1n
π2=nr/V
π2= T^-1/LT^-1
π2=[1]
π1=f(π2)
E/ρV^3= f(nr/V)
Therefore,
The general form of the expression for E in terms of the other variables is
E=ρV^3f(nr/V)