Average of a Velocity Function

[mep12ah]

Homework Statement

Given:

V(n, t, x') = sin [k x' + A (n, t)]

where
k is a constant
n stands for the nth realisation
A is uniformly distributed in [0, 2pi] and
x' denotes the position vector

How can I show that the mean of V i.e. <V> and the mean of the squares <V^2> are independent of x' and indeed for any <V^n>?

Homework Equations

I used <V> = [1/(b-a)] int [a,b] V(n, t, x') dx

where <V> denotes the mean

The Attempt at a Solution

I do not completely understand the question. The average over a complete period or integer multiples of the period is zero for a sine function. The arguments do not affect the average but clearly the interval where you take the average does.

The question implies showing <V>, <V^2> .. <V^n> to be independent for ALL x but being given that A is uniformally distributed in [0, 2pi], does this imply that this is the meant domain of integration for the average?

And how can I generalise for all <V^n>?

 

[mfb]

Mean as average over what? t or n? In that case, how does A depend on t or n?
x'? In that case, it is meaningless to ask how that average depends on x'.

 

[mep12ah]

Mean as average over what? t or n? In that case, how does A depend on t or n?
x'? In that case, it is meaningless to ask how that average depends on x'.

This relates to a fluid flow. I am required to show that it is homogeneous which means that the statistical properties of a property measured at x' do not differ when measured at x'' - all statistical moments should be the same everywhere in physical space.

The average is over x.

 

[mfb]

Ok, now I am confused. Where is x in your equation?

If you average over all x' in some volume, this average does not depend on V(x') for a specific x' if this V(x') is finite and not correlated to any other value in the volume.