# Speed of transverse wave motion

**On a stretched string:**, where*T*= tension in the string and*m*= linear density of string (mass per unit length).**In a solid body:**, where*η*= modulus of rigidity and*ρ*= density of the material.

# Speed of longitudinal wave motion

When the solid is in the form of long bar, , where

*Y*= Young’s modulus of material of rod.**In a liquid medium:****In gases:**

Some Important Points

As solids are most elastic while gases least, i.e.,

*E*. So the velocity of sound is maximum in solids and minimum in gases_{S}> E_{L}> E_{G}*v*

_{steel }>

*v*

_{water }>

*v*

_{air}

5000

*m*/*s*> 1500*m*/*s*> 330*m*/*s*As for sound

*v*_{water}>*v*_{air}while for light*v*<_{w}*v*._{A}As for sound

*v*_{water}>*v*_{air}while for light*v*<_{w}*v*._{A}Water is rarer than air for sound and denser for light.

The concept of rarer and denser media for a wave is through the velocity of propagation (and not density). Lesser the velocity, denser is said to be the medium and vice versa.

# Newton’s formula

He assumed that when sound propagates through air, temperature remains constant. (i.e., the process is isothermal)

*v*

_{air}=

*K*=

*E*, where

_{θ}= P*E*= isothermal elasticity and

_{θ}*P*= pressure.

By calculation,

*v*_{air}= 279 m/s.However the experimental value of sound in air is 332 m/s which is greater than that given by Newton’s formula.

# Laplace correction

Laplace modified Newton’s formula assuming that the propagation of sound in air is a adiabatic process.

*v*=

= (as

*k*=*E*_{φ}_{ }*=**γ*ρ = adiabatic elasticity)=

*×*279 = 331.3 m/s(*γ*_{Air}= 1.41))**Effect of density**

*v*= ⇒

*v*

**Effect of pressure**

Velocity of sound is independent of the pressure of gas, provided the temperature remains constant. (

*P*∝ ρ, constant)**Effect of temperature**

⇒

When the temperature change is small, then

*v*

_{t}_{ }=

*v*

_{0}(1 + α

*t*)

where

*v*_{0}= velocity of sound at 0°C,*v*= velocity of sound at_{t}*t*°C, α = temperature coefficient of velocity of sound.Value of α = 0.608 = 0.61 (approx)

Temperature coefficient of velocity of sound is defined as the change in the velocity of sound, when temperature changes by 1°C.

**Effect of humidity**With increase in humidity, density of air decreases. So with rise in humidity velocity of sound increases. This is why sound travels faster in humid air (rainy season) than in dry air (summer) at the same temperature.

- Sound of any frequency or wavelength travels through a given medium with the same velocity. (
*v*= constant) For a given medium velocity remains constant. All other factors like phase, loudness pitch, quality etc., have practically no effect on sound velocity. - Relation between velocity of sound and root mean square velocity.
*v*_{sound}= and*v*_{rms}=*v*_{sound}= [*γ*/3]^{1/2 }*v*_{rms}