Physics > Motion of Waves > 4.0 Speed of a transverse wave on a string
Motion of Waves
1.0 Introduction
2.0 Mechanical waves
2.1 Transverse waves
2.2 Longitudinal waves
2.3 Differences between transverse and longitudinal waves
3.0 Properties of wave motion
3.1 General equation of wave motion
3.2 Wave function
3.3 Equation of a plane progressive harmonic wave
3.4 Important relations
4.0 Speed of a transverse wave on a string
5.0 Energy associated with a wave
6.0 Questions
4.1 Speed of different types of wave
2.2 Longitudinal waves
2.3 Differences between transverse and longitudinal waves
3.2 Wave function
3.3 Equation of a plane progressive harmonic wave
3.4 Important relations
- As we know, $$\mu = \frac{{mA}}{{lA}} = \frac{{mA}}{V} = \rho A$$ So, the speed of a wave on a stretched string becomes, $$v = \sqrt {\frac{T}{{\rho A}}} $$
where, $\rho $: Density of the string.
- Speed of a transverse wave in a solid is given by, $$v = \sqrt {\frac{\eta }{\rho }} $$
where,
$\eta $: Modulus of rigidity
$\rho $: Density of the solid
- Speed of longitudinal wave through a fluid (liquid or gas) is given by, $$v = \sqrt {\frac{B}{\rho }} $$
where,
$B$: Bulk modulus of the fluid (liquid or gas)
$\rho $: Density of the fluid (liquid or gas)
- Speed of longitudinal wave in a metallic bar is given by, $$v = \sqrt {\frac{Y}{\rho }} $$
where,
$Y$: Young's modulus
$\rho $: Density of the metallic bar
- Speed of longitudinal wave in a medium is given by, $$v = \sqrt {\frac{E}{\rho }} $$
where,
$E$: Modulus of elasticity
$\rho $: Density of the medium
- Newton's formula: Newton assumed that propagation of sound wave in gas is an isothermal process. Therefore, according to Newton speed of sound in a gas is given by, $$v = \sqrt {\frac{P}{\rho }} $$
where,
$P$: Pressure of the gas
$\rho $: Density of the gas
- According to the Newton's formula, the speed of sound in air at N.T.P. is 280 $m/s$. But the experimental value of the speed of sound in air at N.T.P. is 332 $m/s$. Newton could not explain this large difference. Newton's formula was corrected by Laplace.
- Laplace's equation: Laplace assumed that propagation of sound wave in gas is an adiabatic process. Therefore, according to Laplace, speed of sound in a gas is given by, $$v = \sqrt {\frac{{\gamma P}}{\rho }} $$
where, $\gamma $: Adiabatic constant
- According to Laplace's correction the speed of sound in air at N.T.P. is $331.3$ $m/s$. This value agrees fairly well with the experimental values of the speed of sound in air at N.T.P.
- Speed of sound in a gas, $$v = \sqrt {\frac{\gamma }{3}} {v_{rms}}$$
- Factors affecting the speed of sound in a gaseous medium,
Effect of temperature: Speed of sound in a gas is directly proportional to the square root of its absolute temperature. $$v = \sqrt T $$ Speed of sound in a gas at $$t^\circ C$$ is given by, $${v_t} = {v_0}\left[ {1 + \frac{t}{{546}}} \right]$$ or $${v_t} = {v_0} + 0.61t\quad m/s$$ where ${v_0}=332\ m/s$: Speed of sound in the air at $0^\circ C$.
Speed of sound in air increases by 0.61 $m/s$ for every $1^\circ C$ rise in temperature.
Effect of pressure: The speed of sound in a gas is given by, $$v = \sqrt {\frac{{\gamma P}}{\rho }} = \sqrt {\frac{{\gamma RT}}{M}} $$ Speed of sound in gas is independent of the pressure of the gas, provided temperature remains constant.
Effect of humidity: As the humidity increases, the density of gas decreases. So, $$v = \sqrt {\frac{{\gamma P}}{\rho }} $$ Therefore, with rise in humidity speed of sound increases. Since the density of moist air is less than that of dry air, so the speed of sound in moist air is more than the speed of sound in dry air.
Effect of wind: If the wind is blowing, the speed of sound changes. The speed of sound is increased if the wind is blowing in the direction of the propagation of sound wave. But if the wind is blowing opposite to the direction of propagation, the speed of sound is decreased.
