My Engineering Mathematics teacher have a very novel method of teaching. He thinks that while it's all good and well to learn the analytical side of math, he stresses more on the theory, than the application. Thus he gave us these questions that we must answer without a single equation. It's new and I sure enjoy the concept, but I'm not sure if I got it right. Could someone check my answers for me? Thanks!
1. Describe the solution process of the wave equation. Please use several paragraphs for explaining separate steps.
First, we assume that solution of the wave equation is the product of a function of time and a function of position. If we then take the double derivative as outlined in the modeling process and arrange it so that all terms of time are on one side and all the terms of position are on the other, we realized that both sides must equal to a constant. This is because if it wasn’t equal to a constant, then altering one parameter would only change one side, thus proving their inequality. Taking this into consideration, we now have two ordinary differential equations.
Next, by understanding that there is no displacement at either ends of the “string” upon which the wave propagates, we find solutions for the two functions that we separated the solution into so that they satisfy the boundary conditions. It is important to note that the constant that we said that both sides must equal is a negative. This is in order to avoid a trivial solution. By solving, we then get a general solution where the function of time has eigenfunctions of sine and cosine with varying coefficients and eigenvalues that vary with a variable of natural numbers. The function of position is a sine function with infinitely many solutions that also varies with a variable of natural numbers.
Finally, we plug in the initial conditions into the solution function. By plugging in the initial displacement and solving for the remaining coefficient, we find that the solution becomes the Fourier sine series. Then, if we differentiate the solution function and plug-in the initial velocity condition in the same manner, we can also solve for the value of the other coefficient. Now that we know the coefficient values, we can sum this into a series for the varying natural numbers to obtain the solution to the wave equation.
2. Describe the solution process of the two-dimensional wave equation for a rectangular membrane. Please use several paragraphs for explaining separate steps.
Similar to the wave equation for the string, we have a double partial derivative of time and position, except the PDE for position consists of terms for both the x-direction and y-direction. Also similar to the wave equation, we separate according to functions of time and position, except that the function of position has two variables. If we obtain the PDE as before, we get the regular time function, but we get the Helmholtz equation for the position function. If we undergo a second separation of variables in the same manner, we get separate PDEs for x and y. Note that when both cases of separation, the constant that both sides must equal to must be negative if a trivial answer is to be avoided.
Next, we factor in the boundary condition that all edges of the rectangular membrane have no deflection. Then we get that the function of x-direction and y-direction are both sin functions that vary, independently of each other, according to a series of natural numbers. However, the time function consists once more of sine and cosine eigenfunctions with their own coefficients and eigenvalues that vary with a series of natural numbers.
Finally, while we had a single series for the wave equation on a string, we have a double sum series based on the independently varying natural number series for the function of the x-direction and the y-direction. As before, the values of the coefficients can be acquired by plugging in the initial conditions for position and velocity of the membrane. Thus, by summing the found coefficient values in a double Fourier series with respect to the two variables of natural numbers, we get the solution to the wave equation on a rectangular surface.
3. Describe the solution process of the wave equation for a circular membrane. You do not need to show how the wave equation in polar coordinate is obtained. Make sure not to assume that the solutions are radially symmetric.
Much like in the previous cases, here we start with double derivative of time on one side of the equation and a double derivative of position on the other. However, since this is in radial terms and we can’t assume radial symmetry, on the right side we have a double derivative and single derivative of radius and a double derivative of angle. We separate the variables and note that in that form they all equal to a constant. If a trivial solution is to be avoided, the constant must be negative. Also, the separation of variables must be done twice, as in the rectangular membrane case as the right side of the equation has derivatives of radius and angle.
Then, by implementing the boundary conditions of zero deflection at the edges of the circle, we can solve the three PDEs that we got through the previous step. For the time function, we get the usual sine and cosine eigenfunctions with their respective coefficients eigenvalues that vary with natural numbers. For the angle function, we get a similar function of sine and cosine with their own unique coefficients. The values in the sine and cosine functions also vary with natural numbers. Finally, we get the complex radius function which, when simplified, is a linear combination of the Bessel function of the first and second kind. However, when the radius approaches zero, the Bessel function of the second kind gives impossible answers. Thus, its coefficient is considered to be zero. Thus we just have a Bessel function of the first kind with an eigenvalues varying on natural numbers.
Thus, by summing the product of these functions, for all natural numbers, we get the solution for the wave equation on a circular membrane.
4. Suggest a condition that reduces the above solution to the radially symmetric solution.
Needless to say, the difference between a radially symmetric membrane and a radially unsymmetric one is that in a radially unsymmetric membrane, the vibration varies with angle as well as radius length. Thus, the condition that the double derivative of the solution in respect to the angle is zero would effectively eradicate the existence of the angle function and thus give us a solution to a radially symmetric membrane.