Dimensional Analysis:
The velocity u of deep ocean waves is a function of the wavelength w, the acceleration g, and density p. By assuming that u=k(w^a*g^b*p^c), where k is a dimensionless constant, find the relationship between u,w,g and p.

Question

Dimensional Analysis: 
The velocity u of deep ocean waves is a function of the wavelength w, the acceleration g, and density p. By assuming that u=k(w^a*g^b*p^c), where k is a dimensionless constant, find the relationship between u,w,g and p.

anonymous · Answer

Assume that u has dimensions: $$LT^{-1}$$, w has units of L, g has dimensions $$LT^{-2}$$ and p has dimensions $$ML^{-3}$$

anonymous · Answer

using laws of dimensions 
LT^-1 = (L^a)* {(LT^-2)^b}* {(ML^-3)^c}
now compare the powers of each terms L,T&M to find a,b and c
c=0b=1/2,a=1/2
hence u=k{w^1/2 * g^1/2}

anonymous · Answer

Whoa, that's like, I think it's right.

anonymous · Answer

how did u get b=1/2? i have $$M^{c}L^{a+b-3c}T^{-2b} = M^{0}L^{0}T^{0}$$ and solved for it.

anonymous · Answer

look the RHS must have dimensions of u

anonymous · Answer

could u explain what that means?

anonymous · Answer

ohhh... so a+b-3c = 1 & -2b=-1?

anonymous · Answer

law of dimensioal equality states that each individual term in an equation must have the same dimension so what we do here is that assuming the dimensions of the LHS of the equation is the samem as that of the RHS , the given equation is for determining velocity so both sides must have the dimensions of velocity (u)

anonymous · Answer

yeh you got it right

anonymous · Answer

so what do you think they are asking when they want to find the relationship between u,w,g & p?

anonymous · Answer

we need to find the exact powers of the terms in the equation

anonymous · Answer

for clarification, does this also mean that n=5 (including dimensionless constant k) and rank=4?

anonymous · Answer

i mean rank=3 (MLT)

anonymous · Answer

we are least concerened with k as it is dimensionless , In each equation k is an experimentally  determined constatnt , but the other quantities and their powers can be calculated if we know what all factors are effecting the required quantity

anonymous · Answer

i'm just digressing a bit now. since part of dimensional analysis also talks about finding a maximal set (of linearly independent solutions). So I am trying to apply it to this scenario, in the event where it is necessary to find these two independent vectors. How would I approach it?