Understanding Plane Wave Speed In Evans' PDE Book A Detailed Explanation

Hey guys! Ever found yourself scratching your head over a tricky concept in partial differential equations? You're definitely not alone. Today, we're diving deep into a specific question about plane wave speed, inspired by a discussion around Evans' famous book on PDEs. Let's break down the confusion and get a solid understanding of what's going on.

The Puzzle: Deciphering Plane Wave Speed

So, the main question revolves around understanding the speed of a plane wave as described in Evans' Partial Differential Equations. Specifically, there's some confusion about a particular line (marked in red, as the user pointed out) in the book. The user's intuition suggests that the speed of a certain component should be σ, but the book seems to indicate something different. This discrepancy is what we're going to untangle. Let's get into the details so we can make sure we understand the plane wave speed.

Breaking Down Plane Waves

First things first, let's talk about what plane waves actually are. Imagine dropping a pebble into a calm pond. You see ripples spreading outwards in circles, right? Now, picture those ripples extending infinitely in one direction – that's kind of what a plane wave is like. Mathematically, a plane wave is a solution to a wave equation that has the form:

u(x,t) = f(x ⋅ ξ - σt)

Where:

• u(x,t) represents the wave's displacement at position x and time t.
• x is a vector representing the spatial coordinates.
• ξ (pronounced "ksi") is a unit vector that indicates the direction the wave is traveling.
• σ (sigma) is the wave speed.
• f is an arbitrary function that defines the wave's shape.

Key takeaway: The term x ⋅ ξ represents the projection of the position vector x onto the direction of wave propagation ξ. This means that the wave's value is constant on planes that are perpendicular to ξ. That’s why it’s called a plane wave! Now, the σt term represents how the wave shifts over time. Understanding the relationship between these components is crucial for grasping the overall wave behavior.

The Speed Factor: Digging Deeper

The core of the issue here is the speed at which the plane wave propagates. The user's initial thought was that the speed should simply be σ. While this is partially correct, the subtlety lies in how we interpret the wave's components. Remember, σ represents a characteristic speed derived from the wave equation itself. For instance, in the simple wave equation:

utt = c^2uxx

c would be the wave speed. However, when dealing with more complex scenarios, particularly those involving systems of equations or higher-order equations, the relationship between the characteristic speed and the actual observed speed of a specific component can be more nuanced. Therefore, we need to dig deeper and make sure we consider all the components within the equation.

Evans' Approach: A Closer Look

To truly understand the red line in Evans' book, we need to consider the specific context. Without the exact equation and derivation at hand, let’s think through the general approach Evans might be taking. Often, in PDE analysis, plane wave solutions are used as a tool to understand the behavior of more general solutions. The idea is that by decomposing a complex solution into a superposition of plane waves (think Fourier analysis!), we can analyze each plane wave component individually and then combine their effects. This helps in getting a better understanding of the equation overall.

In this context, the "red line" likely refers to a step where Evans is analyzing a particular component of the plane wave solution. It's possible that this component doesn't travel at the characteristic speed σ directly. Instead, its speed might be a function of σ and other parameters in the system. This could arise, for example, if the original PDE is a system of equations, where the different components are coupled together. In that case, the speed of each component might depend on the interactions between them. Let's consider some real world examples of what this might look like.

Possible Scenarios and Examples

To make this clearer, let's imagine some scenarios where the component speed might differ from σ:

1. Systems of Equations: Consider a system of two coupled wave equations. The solutions might involve combinations of plane waves, each with its own speed. The individual components might then travel at speeds that are related to, but not exactly equal to, the characteristic speeds of the individual equations.
2. Higher-Order Equations: For equations involving higher-order derivatives (e.g., the biharmonic equation), the dispersion relation (the relationship between frequency and wave number) can be more complex. This can lead to different components of the solution traveling at different speeds.
3. Variable Coefficients: If the coefficients in the PDE are not constant, the wave speed can vary spatially. In such cases, the local speed of a plane wave component might depend on the position x.

Without seeing the specific equation in Evans' book, it's hard to pinpoint the exact reason for the discrepancy. However, these scenarios highlight the general idea: the speed of a component of a plane wave solution isn't always a straightforward matter of reading off the coefficient in the wave equation. It often requires careful analysis of the underlying structure of the PDE.

Unpacking the Math: A Step-by-Step Approach

Okay, let's get a bit more technical and think about how we might actually calculate the speed of a plane wave component. The key is to look at how the phase of the wave changes over time. Remember our general form for a plane wave:

u(x,t) = f(x ⋅ ξ - σt)

The term (x ⋅ ξ - σt) is called the phase of the wave. The phase essentially tells us where we are on the wave's profile. For example, if f is a sine function, the phase determines whether we're at a peak, a trough, or somewhere in between. Now, imagine we're riding along with the wave, staying at a constant phase. This means that the phase (x ⋅ ξ - σt) remains constant as time changes. Mathematically, we can express this as:

x ⋅ ξ - σt = constant

To find the speed, we need to see how the position x changes with time t. Let's differentiate both sides of the equation with respect to t:

d/dt (x ⋅ ξ - σt) = 0

Assuming ξ is a constant vector, we get:

dx/dt ⋅ ξ - σ = 0

Here, dx/dt represents the velocity of a point moving with the constant phase. Let's call this velocity v. So we have:

v ⋅ ξ = σ

Now, the speed we're interested in is the magnitude of the velocity vector v in the direction of propagation ξ. This is given by the projection of v onto ξ, which is exactly what v ⋅ ξ represents! So, we've arrived back at the conclusion that the speed in the direction of propagation is indeed σ. However, this is where the subtlety comes in. If we're looking at a component of a more complex wave, the actual observed speed might be different.

Back to Evans: Connecting the Dots

Let’s try to connect this back to the red line in Evans' book. Suppose the solution involves multiple plane waves, each with its own amplitude and direction. Then, a particular component might be described by something like:

u_component(x,t) = A(x,t) * f(x ⋅ ξ - σt)

Where A(x,t) is a time-varying amplitude. In this case, the speed of the component would not just be determined by σ, but also by how A(x,t) changes with time and position. This is just one possibility, of course, but it illustrates the kind of thinking that's often required when dealing with PDEs.

The Importance of Context

It's essential to remember that the speed of a plane wave component can be influenced by various factors, including:

• The specific form of the PDE: Is it a simple wave equation, a system of equations, or a higher-order equation?
• The boundary conditions: How do the boundaries of the domain affect the wave's propagation?
• The presence of other waves: Are there interactions between different wave components?

Therefore, when you encounter a question about wave speed, always consider the context carefully. Don't just focus on the characteristic speed σ; think about how the other factors might play a role. The user is right that the specific wave speed can have several factors. We have to consider the PDE, boundary conditions, and the wave functions overall.

Key Takeaways and Next Steps

Okay, guys, let's wrap up what we've covered. The confusion surrounding the red line in Evans' book highlights a crucial point about plane waves: the speed of a component of a plane wave solution isn't always the same as the characteristic speed derived directly from the wave equation. It can be influenced by the system's complexity, interactions between components, and other factors. To truly understand the speed, we need to analyze the phase of the wave and consider how it changes over time and space.

What to do next?

1. Revisit Evans: Go back to the specific section in Evans' book where the red line appears. Try to work through the derivation step by step, paying close attention to how the speed of the component is calculated. If you can share the specific equation and context, that would be super helpful!
2. Explore Examples: Look for examples of wave equations where the component speeds differ from the characteristic speed. Systems of coupled wave equations, higher-order equations, and equations with variable coefficients are good places to start. Try looking up these online and make sure they fit your goals.
3. Practice Problems: Work through practice problems that involve calculating plane wave speeds. This will help you solidify your understanding and develop your problem-solving skills. This is a great next step to truly understand the concepts discussed today.
4. Consult Resources: Don't hesitate to consult other resources, such as online forums, textbooks, or research papers. Sometimes, seeing the same concept explained in different ways can help it click. It never hurts to expand on a concept!

Understanding plane wave speed is a foundational concept in PDEs, so it's worth the effort to get it right. Don't be discouraged if it seems tricky at first – with a bit of practice and careful thinking, you'll get there! Keep exploring, keep questioning, and keep learning! You're doing great! Let me know if you have any more questions, and we can dive even deeper. Good luck, and happy studying! 🚀