The acoustic intensity S (defined as power per unit area) of a sound wave propagating thro… — JAMB UTME

The acoustic intensity $S$ (defined as power per unit area) of a sound wave propagating through a medium of density $\rho$ at speed $v$ is given by the empirical relationship $S = k A^x \omega^2 \rho v^w$ , where $A$ is the wave displacement amplitude, $\omega$ is the angular frequency, and $k$ is a dimensionless constant. Using the principles of dimensional analysis, calculate the numerical value of the exponent $x$ .

Answer: 2

Answer

The numerical value of the exponent

x

is 2.

By applying the principle of dimensional homogeneity, the dimensions of intensity

[S] = M T^{-3}

are equated to

[A]^x [\omega]^2 [\rho] [v]^w = M L^{x - 3 + w} T^{-2 - w}

. Equating time exponents yields

-3 = -2 - w \implies w = 1

. Equating length exponents yields

0 = x - 3 + w \implies x = 2

Step-by-Step Solution

Determine the fundamental dimensions of acoustic intensity

S

[S] = \frac{[\text{Power}]}{[\text{Area}]} = \frac{M L^2 T^{-3}}{L^2} = M L^0 T^{-3}

Intensity is defined as power delivered per unit surface area perpendicular to the direction of propagation.

Write the dimensional formulas for all variables in the given equation

S = k A^x \omega^2 \rho^1 v^w

[A] = L

[\omega] = T^{-1}

[\rho] = M L^{-3}

[v] = L T^{-1}

Each physical quantity must be resolved into fundamental SI dimensions of Mass (

M

), Length (

L

), and Time (

T

Formulate the dimensional balance equation

M^1 L^0 T^{-3} = L^x \cdot T^{-2} \cdot M^1 L^{-3} \cdot L^w T^{-w} = M^1 L^{x - 3 + w} T^{-2 - w}

For physical validity, the dimensions on both sides of an equation must be identical (principle of dimensional homogeneity).

Equate the exponents of Time (

T

) to solve for

w

-3 = -2 - w \implies w = 1

The power of

T

on the left side must equal the sum of powers of

T

on the right side.

Equate the exponents of Length (

L

) to find

x

0 = x - 3 + w \implies 0 = x - 3 + 1 \implies x = 2

Substituting

w = 1

into the length exponent balance yields the value of

x

Key Concept

Principle of Dimensional Homogeneity

Estimated Time:2m 0s

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