Question

Prove: If $$ u(x, t)=f(x-c t)+g(x+c t) $$ then \(u_{t t}=c^{2} u_{x x}\)

Short answer

Question: Prove that the function \(u(x, t) = f(x-ct) + g(x+ct)\) satisfies the wave equation \(u_{tt} = c^{2} u_{xx}\), where \(f\) and \(g\) are arbitrary functions and \(c\) is a constant. Answer: We have proved that \(u_{tt} = c^2 u_{xx}\) for \(u(x, t) = f(x-ct) + g(x+ct)\), which means that the function satisfies the wave equation.

Step-by-step solution

Find the first derivatives with respect to \(x\) and \(t\)

To compute the second derivatives, we first need to find the first derivatives \(u_x\) and \(u_t\). We can use the chain rule to compute the derivatives: $$ u_x = \frac{\partial u}{\partial x} = \frac{\partial}{\partial x} \left( f(x-ct) + g(x+ct) \right) = f'(x-ct) + g'(x+ct) $$ $$ u_t = \frac{\partial u}{\partial t} = \frac{\partial}{\partial t} \left( f(x-ct) + g(x+ct) \right) = -cf'(x-ct) + cg'(x+ct) $$

Find the second derivatives with respect to \(x\) and \(t\)

Now, we can compute the second derivatives \(u_{xx}\) and \(u_{tt}\) using the same approach: $$ u_{xx} = \frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x} \left( f'(x-ct) + g'(x+ct) \right) = f''(x-ct) + g''(x+ct) $$ $$ u_{tt} = \frac{\partial^2 u}{\partial t^2} = \frac{\partial}{\partial t} \left( -cf'(x-ct) + cg'(x+ct) \right) = c^2f''(x-ct) + c^2g''(x+ct) $$

Prove \(u_{tt} = c^2 u_{xx}\)

Now that we have the expressions for \(u_{xx}\) and \(u_{tt}\), we just have to check if they are equal: $$ u_{tt} = c^2f''(x-ct) + c^2g''(x+ct) $$ $$ c^2 u_{xx} = c^2\left(f''(x-ct) + g''(x+ct)\right) $$ We can see that \(u_{tt}\) and \(c^2u_{xx}\) are indeed equal: $$ u_{tt} = c^2 u_{xx} $$ Hence, we have proved the requested property.

Key concepts

Partial Derivatives

In understanding wave equations, it's crucial to grasp the concept of partial derivatives. Think of a multivariable function – a surface with undulations, dips, and peaks. The partial derivative is like focusing your attention on a single coordinate direction and finding the slope in just that direction, while ignoring all others. It's like asking the question, 'If I move along the x-axis only, how quickly is my function changing?'

For the wave equation, when we take the partial derivative of a function like \( u(x, t) = f(x-ct) + g(x+ct) \) with respect to \( x \), we are effectively holding time \( t \) constant and exploring how \( u \) changes spatially. The resulting first derivative, \( u_x \), provides the rate of change along spatial direction, which is a snapshot at a given moment in time of how the wave's shape is transforming.

Second Derivatives

Moving from partial derivatives to second derivatives is shifting from a single still photograph to a slow-motion capture of how a wave's steepness evolves over space or time. Mathematically, second derivatives measure the curvature or the rate of change of the rate of change. In the context of our wave equation, \( u_{xx} \) and \( u_{tt} \) signify the curvature of our 'wave surface' in space and time, respectively.

When we calculate \( u_{xx} \), we're discovering how the slope obtained from the first derivative changes as we move in the x-direction. The same with \( u_{tt} \), it tells us about the acceleration of wave's displacement with respect to time. It's a bit like diagnosing how steeply a wave's profile is curving at any point or how quickly the wave's speed is getting faster or slower with each passing moment.

The Chain Rule

The chain rule is a powerful tool in calculus, like a Swiss Army knife for differentiation. It comes into play when dealing with compositions of functions — functions nested within functions. In our wave equation example, each piece of the function \( u(x, t) \) is a combination of simpler functions that involve both space \( x \) and time \( t \).

The chain rule tells us how to differentiate composite functions gracefully. Instead of clumsily tearing them apart, it allows us to unpack the layers methodically. We treat each composite piece individually, find its derivative, and then multiply them in a specific order. In our case, it helps us calculate derivatives like \( u_t \) with respect to time while it's influenced by how the position x changes with time, hence involving the derivative of \( f \) and \( g \) with respect to their inner functions \( (x-ct) \) and \( (x+ct) \), respectively. The chain rule is essential for handling the dynamic aspects of waves that are influenced by both space and time.