Question

A uniform metal wire of density $\rho$, cross sectional area A and length L is stretched with a tension T. The speed of transverse wave in the wire is,

Answer 1

In this question, we will use three equations. These equations will give us relation between velocities, mass, length, density, tension and volume. Substituting these equations and solving for the value of velocity v, we will get the required result. Further we will discuss the basics of transverse wave and wave, for our better understanding.
Formula used:
$v = \sqrt {\dfrac{T}{\mu }}$
$\mu = \dfrac{m}{l}$
$m = \rho \times V$

Complete answer:
As we know that a transverse wave is defined as a wave whose oscillations are perpendicular to the direction of the wave or path of propagation.
Also, we know that the speed of the transverse wave is given by:
$v = \sqrt {\dfrac{T}{\mu }}$
Here,
$\mu = \dfrac{m}{l}$
Here, m is mass and l is the length of the wire.
Further, we already know that:
$m = \rho \times V$
Here, $\rho$is the density and V is the volume.
We have expression for volume as:
$V = A \times l$
Now, by putting the value of volume in above equation of mass, we get:
$m = \rho \times A \times l$
So,
\eqalign{& \mu = \dfrac{{\rho \times A \times l}}{l} \cr & \Rightarrow \mu = \rho A \cr}
Now, when we put this value in the equation (1), we get:
$v = \sqrt {\dfrac{T}{{\rho A}}}$
Therefore, we get the required answer, which gives us the velocity of the transverse wave in the wire.

Additional information:
As we know, a wave is said to be a disturbance that travels through any medium. Also, a wave transports energy from one location to another location without the transportation of matter.
Now, the frequency is defined as the number of waves that pass a fixed point in unit time. Frequency can also be said as the number of cycles or vibrations undergone by an object or particle or wave during one unit of time.
Two waves are said to be coherent if they are moving with the same frequency and have constant phase difference.
Here, the summation or adding or subtraction of all the waves travelling in a particular medium, gives us the required value of superposition of waves.
Phase of a wave gives us the location of a point within a wave cycle of a repetitive waveform. Mostly, the phase differences between two or more waves are important. Also, when two or more sound waves combine, for example- the difference between the given phases of the two waves is important in determining the resulting waveform.

Note:
We should remember that the phase of the wave can be positive or negative depending on the direction of propagation. Also, a sine wave starts from point zero, whereas the cosine wave starts from one. Note that a transverse wave does not require any medium to travel.