7.25 The speed of sound in a gas, c, is a function of the gas pressure, p, and density, r. Determine, with the aid of dimensional analysis, how the velocity is related to the pressure and density. Be careful when you decide on how many reference dimensions are required. Solution assume c=f(P,r) where c is velocity P is pressure r is density hence taking the dimensions of all the values c=LT^-1 P=ML^-1T^-2 r=ML^-3 hence c=P^ar^b hence LT^-1=(( ML^-1T^-2)^a)*(( ML^-3 )^b) hence eqating the units on both the sides for M-> 0=a+b for L-> 1=-a-3b for T-> -1=-2a gives a=1/2 b=-1/2 hence by dimensional analysis we can say that c=(P^1/2)*(r^-1/2) or c=(p/r)^1/2 .

1. 7.25 The speed of sound in a gas, c, is a function of the gas pressure,
p, and density, r. Determine, with the aid of dimensional
analysis, how the velocity is related to the pressure and density. Be
careful when you decide on how many reference dimensions are
required.
Solution
assume c=f(P,r)
where c is velocity
P is pressure
r is density
hence taking the dimensions of all the values
c=LT^-1
P=ML^-1T^-2
r=ML^-3
hence c=P^ar^b
hence LT^-1=(( ML^-1T^-2)^a)*(( ML^-3 )^b)
hence eqating the units on both the sides
for M-> 0=a+b

2. for L-> 1=-a-3b
for T-> -1=-2a gives a=1/2
b=-1/2
hence by dimensional analysis we can say that
c=(P^1/2)*(r^-1/2)
or c=(p/r)^1/2