Question

The following model determines the vibration of a drum, specifically the vibration of the batter head, where the wave equation is solved in polar coordinates.

1 д? 1ди д?u дr2 + с? д?t т де

u(r,t) is the height of a drum's surface at radius (r) and time (t), which is relative to its position at rest.

c is a parameter which depends on surface density (kg//m³) of the drum head, and with tension t (newtons) applied to it.

Q1. Non-dimensionalise the following equation

Q2. From the question, what dimension should c have?

Answer 1

I du an = tl du (² at? dr² ror 2 has er dimensions of [t uw height Es radies, t time parameter lepends surface density but dimensions are Сэ 2² in has dime of 88² [2] not known. L ge šv +) nu = b řat rar :[]-[] [] 4. [17] = [z'] + [27 [172].[2] CL2=2]; c-[15] it has dimensions of Speed. @ to 2 e2= are at² du I de + rar A I Då dt² 그 (3) + +61) Tarla This side hos no dimensions. iie. [mo iT] is dimensionless Now. This equ nation One can check us eq. of a A follows :- let's take left hand side of <=[+ [ 2 ] + [ - = =D it's dimension -lees. [+] + [t = = [Romos So, CĄ is Non-dimensionalized.