Answer:
a
e(k) = \frac{2a}{c} * sin (\frac{k*a}{2} )
b
G_{v} = \frac{d e(k ) }{dk } = \frac{a^2}{c} * cos (\frac{k* a}{2} )
Explanation:
From the question we are told that
The velocity of transverse waves in a crystal of atomic separation is

Generally the dispersion relation is mathematically represented as

=> 
=> 
=> 
Generally the group velocity is mathematically represented as

A sound wave is the answer
Explanation:
The given data is as follows.
Fluid is water so, density 
Weight flow rate = 500 lbf/s = 2224.11 N/sec
Cross-sectional area (A) = 
= 0.05184 
Hence, weight flow rate will be given as follows.
w = 
2224.11 N/sec = 
V =
m/s
= 4.373 m/s
Thus, we can conclude that average velocity in the given case is 4.373 m/s.
Gas because liquids and solids volumes don't change from switching containers.