Question

In a liquid with density $900kg/{m^3}$,longitudinal waves with frequency $250Hz$ are found to have wavelength $8.0m$,calculate the bulk modulus of the liquid. (in $10$ to the power of $9$ pascals)

Answer 1

Hint: We understand the value of the density, wavelength, and frequency according to the above equation, so we find the liquid bulk modulus. As we know the basic definition of the bulk modulus, we evaluate the concept and address the Pascal value when locating the liquid bulk modulus.
Useful formula:
Velocity of wave,
$v = f\lambda $
Where,
$f$ is frequency
$\lambda $ is wavelength

Complete step by step procedure:
Given by,
Frequency $f = 250Hz$
Density $\rho = 900kg/{m^3}$
Wavelength $\lambda = 8.0m$
A liquid's bulk modulus is related to its compressibility. It is known as the pressure needed to cause a volume change of a liquid device. the ratio of the change in pressure to the fractional volume compression. As pressure is applied to all surfaces, the bulk modulus is nothing but a numerical constant that is used to calculate and define the elastic properties of a solid or fluid.
Now the velocity of wave,
$v = 250 \times 8.0$
On simplifying,
$v = 2000m/s$
Now,
Longitudinal waves, such as sound, are transmitted by media with velocities that depend on the substance's density and elasticity. velocity of a longitudinal wave in a medium Is given by,
$v = \sqrt {K/\rho } $
Or
Rearranging the given equation,
$K = {v^2}\rho $
Or
$K = {2000^2} \times 900$
On simplifying,
\[K = 3.6 \times {10^9}\,Pa\]

Hence, the bulk modulus of the liquid is $K = 3.6 \times {10^9}\,Pa$.

Note: According to the Sound waves, we know including solids, liquids, and gases, need to pass through a medium. By vibrating the molecules in the matter, the sound waves pass through both of these mediums. The molecules are very closely packed in solids. The sound moves in water about four times faster and farther than in air.