Answer:
Part(a): The mass per unit length =
.
Part(b): The velocity of the traveling transverse wave = 
Part(c): The expression for a transverse wave on the string travelling along positive x-direction is 
Part(d): The expression for a transverse wave on the string travelling along negative x-direction is 
Part(e): The equation of the standing wave is
.
Explanation:
Part(a):
The mass per unit length (
) is given by,

Part(b):
According to the figure, the net force on the element (
) of the string, is the sum of the tension in the string (
) and the restoring force. The x-components of the force of tension will cancel each other, so the net force is equal to the sum of the y-components of the force. To obtain the y-components of the force, at 

and at 

The net force (
) on the string element is given by
![F_{net} = F_{V}^{+} + F_{V}^{-}\\~~~~~~~= F_{T}[ (\dfrac{\partial y}{\partial x})_{x = x_{2}} - (\dfrac{\partial y}{\partial x})_{x = x_{1}}]](https://tex.z-dn.net/?f=F_%7Bnet%7D%20%3D%20F_%7BV%7D%5E%7B%2B%7D%20%2B%20F_%7BV%7D%5E%7B-%7D%5C%5C~~~~~~~%3D%20F_%7BT%7D%5B%20%28%5Cdfrac%7B%5Cpartial%20y%7D%7B%5Cpartial%20x%7D%29_%7Bx%20%3D%20x_%7B2%7D%7D%20-%20%28%5Cdfrac%7B%5Cpartial%20y%7D%7B%5Cpartial%20x%7D%29_%7Bx%20%3D%20x_%7B1%7D%7D%5D)
According to Newton's second law of motion,
, where
is mass per unit length. So,
![&& \Delta x \times \mu \dfrac{\partial^{2}y}{\partial t^{2}} = [(\dfrac{\partial y}{\partial x})_{x = x_{2}} - (\dfrac{\partial y}{\partial x})_{x = x_{1}}] \times F_{T}\\&or,& \dfrac{\mu}{F_{T}} \dfrac{\partial^{2}y}{\partial t^{2}} = \dfrac{[(\dfrac{\partial y}{\partial x})_{x = x_{2}} - (\dfrac{\partial y}{\partial x})_{x = x_{1}}]}{\Delta x} = \dfrac{\partial^{2}y}{\partial x^{2}}](https://tex.z-dn.net/?f=%26%26%20%5CDelta%20x%20%5Ctimes%20%5Cmu%20%20%5Cdfrac%7B%5Cpartial%5E%7B2%7Dy%7D%7B%5Cpartial%20t%5E%7B2%7D%7D%20%3D%20%5B%28%5Cdfrac%7B%5Cpartial%20y%7D%7B%5Cpartial%20x%7D%29_%7Bx%20%3D%20x_%7B2%7D%7D%20-%20%28%5Cdfrac%7B%5Cpartial%20y%7D%7B%5Cpartial%20x%7D%29_%7Bx%20%3D%20x_%7B1%7D%7D%5D%20%5Ctimes%20F_%7BT%7D%5C%5C%26or%2C%26%20%5Cdfrac%7B%5Cmu%7D%7BF_%7BT%7D%7D%20%20%5Cdfrac%7B%5Cpartial%5E%7B2%7Dy%7D%7B%5Cpartial%20t%5E%7B2%7D%7D%20%3D%20%20%5Cdfrac%7B%5B%28%5Cdfrac%7B%5Cpartial%20y%7D%7B%5Cpartial%20x%7D%29_%7Bx%20%3D%20x_%7B2%7D%7D%20-%20%28%5Cdfrac%7B%5Cpartial%20y%7D%7B%5Cpartial%20x%7D%29_%7Bx%20%3D%20x_%7B1%7D%7D%5D%7D%7B%5CDelta%20x%7D%20%3D%20%5Cdfrac%7B%5Cpartial%5E%7B2%7Dy%7D%7B%5Cpartial%20x%5E%7B2%7D%7D)
Comparing the above equation with standard wave equation

the the velocity (
) of the transverse wave is

Part(c):
If the wave having wavelength '
' propagates along positive x-direction with the constant velocity '
', then at any instant of time '
' , then its angular displacement '
' is related to the linear displacement 'x' is written as

Also the distance travelled by the wave at time 't' with constant velocity 'v' is
.
So the wave equation can be written as

where '
' is the wave number
and
is the natural angular frequency.
The wave equation propagating along positive x-direction is given by
.
Part(d):
By the same argument given above The wave equation propagating along negative x-direction is given by

Part(e):
The wave equation propagating along positive x-direction is given by

Similarly, the wave equation propagating along negative x-direction is given by

So the equation of the standing wave on the string created by
and
is the superposition of the waves and is given by
