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3 March, 05:11

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.

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  1. 3 March, 08:47
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    Answer: 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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