The power per unit cross-sectional area, E, transmitted by a sound wave is a function of wave speed, V, medium density, rho, wave amplitude, r, and wave frequency, n. Determine by dimensional analysis the general form of the expression for E in terms of the other variables.

E=ρV^3f(nr/V)

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)

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