Question: (a) In a liquid with density \({\bf{1300}}\;{\bf{kg/}}{{\bf{m}}^{\bf{3}}}\), longitudinal waves with frequency 400 Hz are found to have wavelength 8.00 m. Calculate the bulk modulus of the liquid. (b) A metal bar with a length of 1.50 m has density \({\bf{6400}}\;{\bf{kg/}}{{\bf{m}}^{\bf{3}}}\). Longitudinal sound waves take \({\bf{3}}{\bf{.90}} \times {\bf{1}}{{\bf{0}}^{ - {\bf{4}}}}\;{\bf{s}}\) to travel from one end of the bar to the other. What is Young’s modulus for this metal?

The data given in the question can be listed below as;

• The density of the liquid is,\(\rho = 1300\;{\rm{kg/}}{{\rm{m}}^{\rm{3}}}\) .
• Frequency is,\(f = 400\;{\rm{Hz}}\).

• Wavelength is,\(\lambda = 8.00\;{\rm{m}}\)

Longitudinal sound waves are a type of sound waves in which the vibration/oscillation of the particles is in the direction of the propagation of the moving wave in the medium they are traveling.

The formula that we can use to calculate the bulk modulus of the liquid is given as,

\(B = {\lambda ^2}{f^2}\rho \)

Substitute the values in the above expression, and we get,

\(\begin{array}{c}B = {\left( {8.00\;{\rm{m}}} \right)^2}{\left( {400\;{\rm{Hz}}} \right)^2} \times 1300\;{\rm{kg/}}{{\rm{m}}^{\rm{3}}}\\ = 1.331 \times {10^{10}} \cdot \left( {1\;{{\rm{m}}^2} \times 1\;{\rm{H}}{{\rm{z}}^2} \times 1\;{\rm{kg/}}{{\rm{m}}^{\rm{3}}} \times \frac{{1\;{\rm{Pa}}}}{{1\;{\rm{kg}} \cdot {{\rm{s}}^{\rm{2}}}{\rm{/m}}}} \times \frac{{1\;{{\rm{s}}^{\rm{2}}}}}{{1\;{\rm{H}}{{\rm{z}}^2}}}} \right)\\ = 1.331 \times {10^{10}} \cdot \left( {1\;{\rm{Pa}}} \right)\\ = 1.331 \times {10^{10}}\;{\rm{Pa}}\end{array}\)

Thus, the bulk modulus of liquid is \(1.331 \times {10^{10}}\;{\rm{Pa}}\) .