Question

2- The wave equation describing the transverse vibrations of a stretched membrane under tension T and having a uniform surface density p is at2 Find a separable solution appropriate to a membrane stretched on a frame of length a and width b, showing that the natural angular frequencies of such a membrane are given by where n and m are any positive integers.

Answer 1

"Given stretched membrane under tension = T uniform surface density = rho T(partial differential^2 u/partial differential x^2 + partial differential^2 u/partial differential y^2) = rho partial differential^2 u/partial differential t^2 = 0 - (1) w^2 = pi^2 T/rho (n^2/a^2 + m^2/b^2) - (2) u(x, y, t) = x(x) y(y) rho_r1 x"" yu + xy"" u = rho/T u"" xy x""/x + y""/y = rho/T u""/u x""/x = -lambda_^2 y""/y = -lambda^2_2 \r\n where w^2_1 = lambda^2_1 + lambda^2_2 x"" + lambda^2_1 x = 0 y"" + lambda^2_2 y = 0 M + w^2_1 T/rho H = 0 x = c_1 cos lambda_1 x + c_2 sin lambda_1 x y = c_3 cos lambda_2 x + c_4 sin lambda_2 y x = c_5 cos w_1 squareroot T^3/rho + c_6 sin w_1 squareroot T/rho t By using conviction form u (0, y, t) x(0) = 0 Rightarrow c_1 = 0 u (0, y, t) = 0"