JEE Main & Advanced Physics Wave Mechanics Velocity of Transverse Wave

The velocity of a transverse wave in a stretched string is given by \[v=\sqrt{\frac{T}{m}}\]; where T = Tension in the string;  m = Linear density of string (mass per unit length).

(1) If A is the area of cross-section of the wire then \[m=\rho A\]

\[\Rightarrow \] \[v=\sqrt{\frac{T}{\rho A}}=\sqrt{\frac{S}{\rho }}\]; where S = Stress \[=\frac{T}{A}\]

(2) If string is stretched by some weight then

\[T=Mg\]

\[\Rightarrow \] \[v=\sqrt{\frac{Mg}{m}}\]

(3) If suspended weight is immersed in a liquid of density \[\sigma \] and \[\rho =\] density of material of the suspended load then

\[T=Mg\left( 1-\frac{\sigma }{\rho } \right)\]

\[\Rightarrow \] \[v=\sqrt{\frac{Mg(1-\sigma /\rho )}{m}}\]

(4) If two rigid supports of stretched string are maintained at temperature difference of \[\Delta \theta \]  then due to elasticity of string.

\[T=YA\alpha \Delta \theta \]

\[\Rightarrow \]\[v=\sqrt{\frac{YA\alpha \Delta \theta }{m}}\]

\[=\sqrt{\frac{Y\alpha \Delta \theta }{d}}\]

where Y = Young's modulus of elasticity of string, A = Area of cross section of string, \[\alpha =\] Temperature coefficient of thermal expansion, d = Density of wire \[=\frac{m}{A}\]

(5) In a solid body : \[v\,=\,\sqrt{\frac{\eta }{\rho }}\]

where \[\eta =\] Modulus of rigidity; \[\rho =\]Density of the material.