Sound Waves Concept Page - 3

Law

State Laplace's correction to Newton's Formula for velocity of sound in gases and its assumptions

Velocity of sound is given by: v=γPρ−−−√$v=\sqrt{{\displaystyle \frac{\gamma P}{\rho }}}$

Laplace's assumption of isothermal conditions to prevail when sound travels through air, Newton has applied Boyle's law to pressure changes and volume. In a region of compression, there is nominal increase in temperature and in a region of rarefaction, there is marginal decrease in temperature. These changes in pressure occur rapidly and air is a poor conductor of heat thus, equalization of temperature among the different regions was improbable, according to Laplace. He was of the view that the changes in temperature occur under adiabatic conditions, i.e., no heat enters into the gas from outside or leaves it from inside. The heat developed in the compressed layers remains fully confined to these layers and has no time to get dissipated into the entire body of the gas. Similarly, the cold caused in the rarefied layers cannot be compensated, by flow of heat inside it from other layers. 

Example

Find particle velocity of a given travelling transverse wave

Example: The equation of a progressive wave is Y=asin(ωt−kx)$Y=a\sin (\omega t-kx)$, then find the velocity of the wave.

Solution:
Given equation of wave is y=asin(ωt−kx)$y=asin(\omega t-kx)$
dydt=aωcos(ωt−kx)${\displaystyle \frac{dy}{dt}}=a\omega cos(\omega t-kx)$
dydx=−akcos(ωt−kx)${\displaystyle \frac{dy}{dx}}=-akcos(\omega t-kx)$
Velocity of wave v=dxdt=(dydt)(dydx)=aωak=ωk$v={\displaystyle \frac{dx}{dt}}={\displaystyle \frac{\left({\displaystyle \frac{dy}{dt}}\right)}{\left({\displaystyle \frac{dy}{dx}}\right)}}={\displaystyle \frac{a\omega }{ak}}={\displaystyle \frac{\omega }{k}}$

Formula

State and use the formula for velocity of longitudinal waves in solids

Example:The speed of longitudinal waves in a steel bar is ( Y=2.0×1011Pa$Y=2.0\times {10}^{11}Pa$ and Density =7800kg/m3$=7800kg/m^3$ ).Solution:
The speed of  longitudinal waves in a solid(steel bar) is given by
v=Yρ−−√$v=\sqrt{{\displaystyle \frac{Y}{\rho }}}$
where, Y$,Y$ is the Young's modulus of steel bar and ρ$\rho$ is the density of steel bar.
v=2×10117800−−−−−−−√$v=\sqrt{{\displaystyle \frac{2\times {10}^{11}}{7800}}}$
v=0.002564×1010−−−−−−−−−−−−−√$v=\sqrt{0.002564\times {10}^{10}}$
v=0.05063×105$v=0.05063\times {10}^5$
v=5063 ms−1$v=5063m{s}^{-1}$

Example

Newton's Formula for velocity of sound in gases and with assumptions

Newton's Formula for velocity of sound in gases:
v=Bρ−−√$v=\sqrt{{\displaystyle \frac{B}{\rho }}}$,
where B$B$ is the bulk modulus of elasticity. Newton assumed that the temperature remains constant when sound travels through a gas. Therefore, the process is isothermal which is essentially a slow process and Boyle's law can be applied. At a region of compression, the pressure increases and volume decrease.

Example

Use the relationship between speed of sound in the gas and rms speed of a gas

Example: In a mixture of gases, the average number of degrees of freedom per molecule is 6$6$. The rms speed of the molecules of the gas is c$c$. Find the velocity of sound in the gas.

Solution:
velocity of sound=(γ3)1/2×rms speed of molecules${\left({\displaystyle \frac{\gamma }{3}}\right)}^{{}^1{/}_2}\times rmsspeedofmolecules$
=⎛⎝⎜⎜⎜1+2f3⎞⎠⎟⎟⎟1/2×c$={\left({\displaystyle \frac{1+{\displaystyle \frac{2}{f}}}{3}}\right)}^{{}^1{/}_2}\times c$  (∵γ=1+2f)$(\because \gamma =1+{\displaystyle \frac{2}{f}})$
=⎛⎝⎜⎜1+263⎞⎠⎟⎟1/2×c$={\left({\displaystyle \frac{1+{\displaystyle \frac{2}{6}}}{3}}\right)}^{{}^1{/}_2}\times c$
=(86×3)1/2×c$={\left({\displaystyle \frac{8}{6\times 3}}\right)}^{{}^1{/}_2}\times c$
=(49)1/2×c$={\left({\displaystyle \frac{4}{9}}\right)}^{{}^1{/}_2}\times c$
=23c$={\displaystyle \frac{2}{3}}c$

Definition

Properties of the medium to propagate sound

The medium for propagation of sound should be:

• Elastic so that particles come back to the original position.
• Should have inertia so that particles can store mechanical energy.
• Medium should be frictionless so that there is no loss of energy.

Result

Factors affecting speed of sound

Speed of sound is affected by density, temperature, humidity and direction of wind.

Result

Effect of density on speed of sound

v=γPρ−−−√$v=\sqrt{{\displaystyle \frac{\gamma P}{\rho }}}$
v∝1ρ√$v\propto {\displaystyle \frac{1}{\sqrt{\rho }}}$
Hence, speed of sound decreases with increase in the density of the medium.

Result

Effect of temperature on speed of sound

VT2=VT1+0.61(T2−T1)$V_{T2}=V_{T1}+0.61(T_2-T_1)$
where
VT2:$V_{T2}:$ speed of sound wave at temperature T2$T_2$
VT1:$V_{T1}:$ speed of sound wave at temperature T1$T_1$
Hence, speed of sound increases with increase in temperature.

Result

Effect of humidity on speed of sound

Speed of sound increases in proportion to humidity in air. Humidity has a small but significant effect on speed of sound (causing it to increase by about 0.1%-0.6%), the reason being oxygen and nitrogen molecules of the air are replaced by lighter molecules of water.

The approximate speed of sound in dry (0% humidity) air, in meters per second, at temperatures near 00 C$0^{0}C$, can be calculated from:
 v=(331.3+0.606T)$v=(331.3+0.606T)$ m/s

Result

Effect of direction of wind on speed of sound

The speed of sound increases when the sound wave is moving in the direction of wind.
The speed of sound decreases when the sound wave is moving in the direction opposite to the direction of wind.

Result

Factors not affecting the speed of sound

Factors not affecting the speed of sound are pressure, amplitude, wavelength and frequency of the sound wave.

BookMarks

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