Question

Calculate the period of a sound wave that sustains an intensity of 67w / m2 at an amplitude of 12m considering that its velocity is 34m / s and a fluid density of 89kg / m3.

Answer 1

Let's first find the bulk modulus of the fluid:

$$v=\sqrt{\dfrac{B}{\rho}},$$ $$B=\rho v^2=89\ \dfrac{kg}{m^3}\cdot(34\ \dfrac{m}{s})^2=1.03\cdot10^5\ Pa.$$

Then, we can find the angular frequency of the sound wave from the formula:

$$I=\dfrac{1}{2}\sqrt{\rho B}\omega^2A^2,$$ $$\omega=\sqrt{\dfrac{2I}{\sqrt{\rho B}A^2}},$$ $$\omega=\sqrt{\dfrac{2\cdot67\ \dfrac{W}{m^2}}{\sqrt{89\ \dfrac{kg}{m^3}\cdot 1.03\cdot10^5\ Pa}\cdot(12\ m)^2}}=0.017\ \dfrac{rad}{s}.$$

Finally, we can find the period of the sound wave:

$$T=\dfrac{2\pi}{\omega}=\dfrac{2\pi}{0.017\ \dfrac{rad}{s}}=370\ s.$$