Question

. (40 points: A membrane is stretched under tension r with uniform surface density o. (Small-amplitude) transverse displacements of this stretched membrane satisfy the wave equation కా - ఆ (10) Suppose the membrane is stretched in a rectangular frame lying in the plane z= 0, with side-lengths a, b. mitially stationary at all points, the membrane is struck at t-0 and thus set vibrating- 3.1. Show that e satisfies the following expression: veu)-ΣΣ..-ΣΣ Β.in may sin sin (wt] %3D (11) (you do not need to determine Bm), where w is the natural frequency of vibration of the (n, m)'th mode am, and will be determined shortly. 3.2. You may recall that Boas' discussion of vibrations of a circular membrane involved Bessel functions to describe the vibrations that resulted in that situation. In a sentence or two, indicate the main difference between that situation and the present question that results in no Bessel functions appearing in expression (11). 3.3. With reference to your working for part 3.1, or otherwise, show that the natural frequencies of vibration of the membrane satisfy the expression m2 (12) with n, m both positive integers. 3.4. For each of the following four normal modes {vn. ta. ta2, ta3}, sketch a snapshot of the membran at either extreme of sin(wt), including the nodal lines (ie. lines in z, y for which t(r,y.t)nm =0 for al 1). In cach case, indicate with the "+" and symbols whether a region is above or below the plan 0 (see Figure 3 for an example). 3.5. Suppose the frame lengths are a = 2 feet and b = 4 feet. Are there cases in which two or mon normal modes have the same angular frequency? If yes, provide an example. If no, state why not. t>0, one vibration mode b. Figure 3: Coordinates for question 3. One example normal mode of vibration is indicated.

Answer 1

3.1 and 3.3)

First, let us check if the given function satisfies the boundary condition.

(7'fira

The boundary conditions are no displacement at the boundaries.

thus checking for x = 0,

0 0 10 (0,1, 8) = Žmasin(x 9 sin (1974) sinfusi) = 0 m=1

similarly for y=0,

Χ Χ (α, 0, 0 - ΣΣΒανκίνι (πε)τκ 0].τιμή = 0 n= 1 1-1

for x = a,

X0X0 usta, 4, 8) = { ŽEmusin (974) sin (1974) sin[w] = 0 t=1 =1

for y = b

0= [We u se a ਤੇ = (41)

thus the given satisfies the boundary condition.

(7'fira

now to check if it satisfies the wave equation:

X0 computer , 0, 0) = -ow? E Busin (1974sin(974) sinfurt

and

V?y(x, y,t) = --[immo Bunsin [17:] sin [M74) sinut] n=1 m=1

thus satisfies the wave equation if

(7'fira

$\omega^2 =\frac{\tau\pi^2}{\sigma}\Bigl[\frac{n^2}{a^2}+\frac{m^2}{b^2}\Bigr]$

3.5) for this,

$\omega^2 =\frac{\tau\pi^2}{\sigma}\Bigl[\frac{n^2}{4}+\frac{m^2}{16}\Bigr]$

now
m (m/2)
and
n 4 (2n) 16

thus (n,m) and (m/2,2n) are a pair of modes which have the same frequency.