20.2. Solution by Separation of Variables

We assume a solution of the form:

u(x,y,t)=X(x)Y(y)T(t)

In this section, we start by proposing a specific form for the solution of the two-dimensional wave equation. We express the displacement function, u(x,y,t), as a product of three separate functions: X(x), which depends only on the x-coordinate; Y(y), which depends only on the y-coordinate; and T(t), which depends only on time. This method is known as 'separation of variables.' By assuming such a form, we can simplify the problem into smaller, more manageable equations that can be solved individually.

Think of a musician playing a stringed instrument, where the sound can be broken down into different components: the vibration of the string (X), the way the instrument body resonates (Y), and the timing of the strumming or plucking (T). Just like how each part contributes to the overall sound, each function in our solution contributes to the overall displacement of the membrane.

Substituting into the wave equation:

d²T (cid:18) d²X d²Y (cid:19) = c²T Y + X dt² dx² dy²

After assuming the solution form, we substitute it into the two-dimensional wave equation. This substitution results in an equation that relates the individual components of time and space. The left side corresponds to the time dependence, while the right side relates to the spatial functions. This sets us up for the next step where we will separate the variables further to isolate terms depending only on time from those depending on space.

Imagine you have a jigsaw puzzle, and you start placing pieces into a framework; each piece fits into one specific spot, leading to a clearer picture. Similarly, by substituting our assumed solution into the wave equation, each function starts to find its place, revealing the overall dynamic behavior of the membrane.

Dividing both sides by XYT:

1 d²T (cid:18) 1 d²X 1 d²Y (cid:19) = c² + T dt² X dx² Y dy²

Since the left side depends only on t and the right side only on x and y, both must equal a constant, say −λ.

Next, we divide the entire equation by the product XYT to isolate the variables completely. This separation results in an equation where the left side depends entirely on T (time) and the right side only on X (space in x-direction) and Y (space in y-direction). To balance the equation, we set both sides equal to a constant, which we denote as −λ. This step leads us towards establishing relationships between the spatial and temporal components, leading to the formation of ordinary differential equations for each variable.

Picture cooking in a kitchen with different ingredients. To get a well-balanced dish, you separate the ingredients according to their type and then measure each one precisely to ensure perfect ratios. Similarly, here, we are separating time and space components to cleanly and accurately analyze their individual effects on the membrane's vibration.

Further separation gives:

1 d²X 1 d²Y =−α², =−β², so that λ=α²+β² X dx² Y dy²

In this step, we further separate the equations into distinct parts: one for the function X and another for Y. We express these relationships as second-order ordinary differential equations. By introducing constants α² and β², we relate them back to our original constant λ: λ = α² + β². This indicates that the spatial components are now defined by these new constants, setting the stage for solving these resulting differential equations individually.

Think of how you might segment a complex task into simpler, digestible parts. A student might break down studying for an exam into smaller sections — one for math, another for science, and so forth. Each section gets its attention and methods, just as we are breaking down the vibration problem into equations that can each be solved step-by-step.

Then the time equation becomes:

d²T + c²(α²+β²)T =0 dt²

After separating the spatial variables, we derive the equation for the time-dependent part (T). It comes out to be a second-order ordinary differential equation that includes the constant terms α² and β², along with a wave speed factor, c. This form implies harmonic motion, leading to solutions involving sine and cosine functions, which are essential in modeling vibrational behavior.

A pendulum swinging back and forth exhibits simple harmonic motion. If we analyze its movement over time, we can represent its position as a function of time in terms of sine and cosine. Similarly, our equation implies that the displacement of the membrane over time will also exhibit periodic behavior, characterized by these functions.

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