Question

Find the deflection of the membrane of sides a and b with c ^ 2

= 1 for the initial deflection f(x, y) = sin (6pi*x)/a * sin
(2pi*y)/b and initial velocity

Answer 1

To find the deflection of the membrane with the given initial deflection and initial velocity, you'll need to solve the wave equation, which describes the motion of the membrane. The wave equation is:

• 
  `\partial^2u/\partial t^2 = c^2(\partial^2u/\partial x^2 + \partial^2u/\partial y^2)`

Where:

• u(x, y, t) is the deflection of the membrane at position (x, y) and time t.
• c is the wave speed (in this case, c^2 = 1).
• a and b are the sides of the membrane.

Given the initial deflection f(x, y) = sin(6
`\pi`x/a) * sin(2
`\pi`/b) and initial velocity, you'll need to set up and solve this partial differential equation with appropriate initial conditions. Solving this equation analytically can be complex and might require separation of variables or Fourier series methods depending on the boundary conditions.

Without specific boundary conditions, it's challenging to provide an exact solution. You may need to use numerical methods or specialized software to solve this partial differential equation with the given initial conditions and any additional boundary conditions.