Answer
(a) $v = \sqrt{\frac{B}{\rho}}$
The equation gives the speed of sound in units of m/s
(b) $v = C\sqrt{\frac{B}{\rho}}$
No other combination of $B$ and $\rho$ can give the dimensions of speed, so the given equation must be correct.
Work Step by Step
(a) Speed $v$ is measured in units of $m~s^{-1}$
$B$ is measured in units of $N/m^2$ which can be expressed as $kg~m^{-1}~s^{-2}$
$\rho$ is measured in units of $kg~m^{-3}$
$v = \sqrt{\frac{B}{\rho}}$
We can consider the units:
$\sqrt{\frac{N/m^2}{kg/m^3}} = \sqrt{\frac{kg/m~s^2}{kg/m^3}} = \sqrt{\frac{m^2}{s^2}} = m/s$
The equation gives the speed of sound in units of m/s
(b) Let's assume that $v = C~B^a~\rho^b$, where $C$ is a dimensionless constant.
Then: $m~s^{-1} = (kg~m^{-1}~s^{-2})^a~(kg~m^{-3})^b$
We can consider the units of $s$:
$(s^{-2})^a = s^{-1}$
$s^{-2a} = s^{-1}$
$-2a = -1$
$a = \frac{1}{2}$
We can consider the units of $m$:
$(m^{-1})^a~(m^{-3})^b = m^1$
$-a-3b = 1$
$3b = -a-1$
$3b = -\frac{1}{2}-1$
$3b = -\frac{3}{2}$
$b = -\frac{1}{2}$
We can use the exponents to write the equation:
$v = C~B^a~\rho^b$
$v = C~B^{1/2}~\rho^{-1/2}$
$v = C\sqrt{\frac{B}{\rho}}$
No other combination of $B$ and $\rho$ can give the dimensions of speed, so the given equation must be correct.
