Speed of waves: Water

Question:
The bulk modulus of water is 2.04$\times$10$^9$ Pa. The frequency of a water wave is found to be 262 Hz. What is the wavelength?

Answer:
The density of water is 1000 kg/m$^3$. The speed of the wave is given by
\[
v = \sqrt{\frac{B}{\rho}}
\]
where $B$ and $\rho$ are the bulk modulus and the density, respectively. Plug in the numbers.
\[
v =  \sqrt{\frac{2.04\times 10^9}{1000}}=1428 \ \mathrm{m/s}
\]
We know the relationship:
\[
v = f\lambda
\]
Therefore, the wavelength is
\[
\lambda = \frac{v}{f}=\frac{1428}{262}=5.45 \ \mathrm{m}
\]

Powered by Hirophysics.com

Sound intensity

Question:
There is a speaker emitting sound with a uniform power of 250 W. At what distance will the intensity be just below the threshold of pain, which is 1.00 W/m$^2$?

Answer:
The sound intensity is given by
\[ I = \frac{P}{A} \]
The sound emitted uniformly in space, so it is spherically propagated. The area of sphere is $4\pi r^2$. Thus,
\[ I=\frac{P}{4\pi r^2} \]
Solve for the distance.
\[ r^2 = \frac{P}{4\pi I} \\
    r = \sqrt{\frac{P}{4\pi I}}\]
Plug in the numbers.
\[ r = \sqrt{\frac{250 (\mathrm{W})}{4\pi \times 1.00 (\mathrm{W/m}^2)}} \]
Therefore, the distance is
\[ r = 4.46 \mathrm{m}\]

Powered by Hirophysics.com