Sound Waves

Sound is a mechanical wave which can be produced with a medium – solid, liquid, or gas. It transfers energy through the medium. The speed of sound depends on the type of medium where it travels. Sound waves travel fastest in solids, faster in liquids and travel very slow in gases.

The properties of medium that affect the speed of sound include temperature, density, and elasticity of the material. Sound travels quickly in material with higher elastic properties. According to density, sound travels slowly in denser materials. As in temperature, sound travels faster at high temperature and slower at low temperature.

The following equations are used to solve the speed of sound depending on the medium where it travels.

Speed of sound in air: $$v=331\;m/s+\frac{0.6\;m/s}{^\circ C}(T)$$

Speed of sound in solids: $$v=\sqrt{Y}$$

Speed of sound in liquids and gases: $$v=\sqrt {\frac{\beta}{\rho}}$$

where

$$v=\text{speed of sound}\\ Y=\text{Young’s modulus of elasticity in }N/m^2\\ \beta=\text{bulk modulus of elasticity in }kg/m^3\\ \rho=\text{density of medium}$$

Example 1

It was a cold night with about $$19 \;^\circ C$$ of temperature when a mother heard his son crying because of stomachache. If the room of the son from his mother’s is $$5 \;m$$ away, how much time does it take for the mother to hear his son’s cry?

Given:

$$T=19\;^\circ C\\ t=5 \;s$$

Solution 1:

Solve for the speed of sound in air.

$$v=331\;m/s+\frac{0.6\;m/s}{C}(19^\circ C)\\ \quad=331\;m/s + 11.4\;m/s\\ \quad=342.4\; m/s$$

Solution 2:

The time is calculated using the equation $$v=\frac dt$$.

$$t=\frac dv=\frac {5\;m}{342.4\;m/s}=0.015\;s$$

Hence, the mother heard his son cried after $$0.015 \;s$$.

Example 2

Solve for the bulk modulus of the seawater given that the speed of sound waves in seawater is $$1,522\;m/s$$ and the density of seawater is $$1.027\;g/cc$$.

Given:

$$v= 1, 522\;m/s\\ \rho=1.027\;g/cc$$

Solution 1:

Convert first the density to $$kg/m^3$$.

$$\rho=\frac{1.027\;g}{cc} \times \frac {1000\;kg}{m^3}=1,027\;kg/m^3$$

Solution 2:

Using the formula $$v=\sqrt {\frac{\beta}{\rho}}$$, the equation of the bulk modulus is expressed as

$$v=\sqrt {\frac{\beta}{\rho}}$$     ; squaring both sides

$$v^2= \begin{pmatrix} \sqrt {\frac{\beta}{\rho}} \end{pmatrix}^2$$

$$v^2=\frac {\beta}{\rho}$$

hence, the equation for $$\beta$$ is,

$$\beta=v^2 \rho\\ \;\;=(1,522\;m/s)^2 (1,027\;kg/m^3)\\ \;\;=2.38 \times 10^9\;N/m^2$$

The bulk modulus is equal to $$2.38 \times 10^9\;N/m^2$$.