In the study ofshallow water waves, it turns out that under certain conditions,t
ID: 1815477 • Letter: I
Question
In the study ofshallow water waves, it turns out that under certain conditions,the critical water depth associated with the behavior of the wavesis a function of surface tension, density, and gravitationalconstant, i.e., hc =f( s, , g).(a) Show workusing Dimensional Analysis to find the functional relationshipbetween these parameters.
(b) If the surface tension were to increase by a factor of 2, whatwould happen to hc ? Give your answer insentence form.
Thank you! Please help with all or some of the problem! Thanks again! In the study ofshallow water waves, it turns out that under certain conditions,the critical water depth associated with the behavior of the wavesis a function of surface tension, density, and gravitationalconstant, i.e., hc =f( s, , g).
(a) Show workusing Dimensional Analysis to find the functional relationshipbetween these parameters.
(b) If the surface tension were to increase by a factor of 2, whatwould happen to hc ? Give your answer insentence form.
Thank you! Please help with all or some of the problem! Thanks again! (a) Show workusing Dimensional Analysis to find the functional relationshipbetween these parameters.
(b) If the surface tension were to increase by a factor of 2, whatwould happen to hc ? Give your answer insentence form.
Thank you! Please help with all or some of the problem! Thanks again!
Explanation / Answer
Note that my convention is Mass[M], Time[Th], Temperature[T],Length[L]h [=] L
s [=] Energy/area [=] (M L^2 / Th^2)/(L^2) [=] M/Th^2 p [=] Mass/volume [=] M/L^3 g [=] length/time squared [=] L/Th^2
# groups = #vars - #dimensions = 4-3 = 1
we start with h = (s^a) (p^b) (g^c) [=] L, and we are lookingfor a, b and c
You can start by seeing that both s and p have an M term inthe numerator, so, in order to cacel, one must be in the numeratorof the final number and the other in the denominator, and both musthave the same exponent. In other words, b = -a
So we have h = (s/p)^a (g^c) [=] L
We also need to cancel the Th^-2 in s and g, so, likewise, c =-a
So we have h = (s/ (p g))^a [=] L
No, we need the entire think to have dimensions of L -- if youplug in the dimensions now, you get: L = ((M/Th^2)(L^3/M)(Th^2/L))^a = (L^2)^a, so a is 1/2
Therefore h is a function of (s/(p g))^(1/2)
This can also tell us that if we doublethe surface tension, h willincrease by a factor of 2^(1/2), or 1.414
Hope that helps s [=] Energy/area [=] (M L^2 / Th^2)/(L^2) [=] M/Th^2 p [=] Mass/volume [=] M/L^3 g [=] length/time squared [=] L/Th^2
# groups = #vars - #dimensions = 4-3 = 1
we start with h = (s^a) (p^b) (g^c) [=] L, and we are lookingfor a, b and c
You can start by seeing that both s and p have an M term inthe numerator, so, in order to cacel, one must be in the numeratorof the final number and the other in the denominator, and both musthave the same exponent. In other words, b = -a
So we have h = (s/p)^a (g^c) [=] L
We also need to cancel the Th^-2 in s and g, so, likewise, c =-a
So we have h = (s/ (p g))^a [=] L
No, we need the entire think to have dimensions of L -- if youplug in the dimensions now, you get: L = ((M/Th^2)(L^3/M)(Th^2/L))^a = (L^2)^a, so a is 1/2
Therefore h is a function of (s/(p g))^(1/2)
This can also tell us that if we doublethe surface tension, h willincrease by a factor of 2^(1/2), or 1.414
Hope that helps