10.7. Solving PDEs Using Fourier Sine Transform

Now consider problems where the solution vanishes at the boundary x = 0, which is ideal for Fourier Sine Transform.

This chunk introduces the context where Fourier Sine Transform (FST) is effectively applied. In many physical situations, especially in mechanical and civil engineering, we encounter problems where a function approaches zero at the boundary (here, at x = 0). This feature of vanishing conditions at the boundaries makes Fourier Sine Transform the appropriate tool, as it is specifically designed to handle such cases. Thus, if we expect our solution to decay to zero at the boundaries, we would opt for FST over other transforms.

Imagine holding one end of a rope tightly (this represents the boundary at x = 0) while the other end dangles freely. The vibrations or movements you create start at the fixed end but gradually diminish as they travel along the rope towards the free end. In this scenario, the Fourier Sine Transform effectively models how the vibrations are controlled from the boundary condition to the end of the rope, where they eventually disappear.

Let us consider the one-dimensional wave equation: ∂²u/∂t² = c² ∂²u/∂x² With boundary conditions: u(0,t)=0, lim (x→∞) u(x,t)=0, u(x,0)=f(x), ∂u/∂t |_(t=0)=0

In this chunk, we are presented with the one-dimensional wave equation, which describes how wave-like phenomena propagate through a medium. The equation itself includes second derivatives with respect to both time (t) and space (x), meaning it captures the acceleration of wave displacement over time and the spatial characteristics of the wave. The boundary conditions specify that at the fixed location x = 0, the displacement must always be zero, and as we move further away (toward infinity), the displacement should approach zero. The initial conditions outline that we've started with an initial shape of the wave defined by f(x) and that the initial velocity of the wave is zero.

Think of a stretched membrane, like a drum skin, where striking the drum creates waves that ripple outwards. The point where you strike (where the drum is fixed) represents the boundary condition (u(0,t) = 0), where there's no displacement of the membrane at the rim, while the waves themselves diminish as they travel further from the source, consistent with the condition that the wave height approaches zero at infinity.

Apply the Fourier Sine Transform: ∂²U/∂t² = -c²s²U This is a second-order ODE in t: U(s,t)=A(s)cos(cst)+B(s)sin(cst)

Once we apply the Fourier Sine Transform to the wave equation, we end up with a transformed equation that represents the same physical phenomenon in the frequency domain. This transformed equation is a second-order ordinary differential equation (ODE) in terms of time t. The general solution to this ODE is expressed in terms of sine and cosine functions, where A(s) and B(s) are coefficients that depend on the transformed initial conditions. The cosine term represents the oscillations of the wave associated with the initial shape, while the sine term would account for any initial velocities.

Returning to our drum example, if one were to tap the drum softly, it would create a sound wave characterized by specific frequencies (higher and lower notes). The sine and cosine functions represent these frequencies, capturing the essence of how the drum vibrates over time, thus allowing us to predict how the sound changes shortly after you hit it based on the initial conditions.

Initial conditions give: U(s,0)=F{s{f(x)}}=A(s), ∂U/∂t|_(t=0)=0⇒B(s)=0 Thus, U(s,t)=F{f(x)}·cos(cst) ⇒u(x,t)=F{f(x)}cos(cst)sin(sx)ds

This chunk illustrates how the coefficients obtained from the initial conditions help refine our solution for U(s,t). In this case, since the initial velocity of the wave is set to zero, it simplifies the solution by eliminating the B(s) term. Thus, the solution U(s,t) depends solely on the initial shape of the wave and oscillates with cosine over time. We then perform the inverse transform to get the actual displacement function u(x,t) in terms of the original function f(x) and sine terms, encapsulating how the wave travels through the medium over time.

Imagine being at a concert. The musicians start playing a familiar tune (f(x) is the orchestra’s performance). Half a second in, there's absolute silence at the start (initial velocity is zero), resulting in a smooth wave pattern of sound traveling outwards to the audience. The final displacement function reflects both the initial tune played and the nature of how sound waves propagate through the air (represented by the sine terms, affecting where the sound is heard over time).

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