Mathematical Wave: Solving the Wave Propagation Equation (1.94)

[hanson]

Hi all.
I am learning some mathematics about wave.
This thread is rather long becaue I find it really difficult for me to comprehend the stuff completely. I have scanned 3 pages of the book I am reading. I REALLY hope you guys won't mind reading my long thread and the 3 pages and lend me your HELPING HAND. I am really frustrted now...

The example is illustrating the ues of asymptotic expansion to solve the wave propagation equtaion (1.94).

First of all, in page 39, under euqation (1.95), it states that we are seeking a solution for x=O(1) and t=O(1). How should I interpret this? Apparetly, it means x is bounded by some constant, right? But why it says later that we would solve the equation in the domain -infinity < x < +infinity?
Does x=O(1) only means x is bounded by some constant as epsilon approaches o, but it doesn't follow that x is bounded itself?

Secondly, in page 40 bottom, it says "For f(x) on compact support...Further, for our stated condition on f(x), we need consider only zeta = O(1)..."
What kind of "stated condition" is it referring to? And why we need to consider only zeta = O(1)?

Even I accept that we only need to consider zeta = O(1), on page 41, second paragraph, why for zeta = x-t = O(1), then t = O(epsilon ^-1) implies that x = O(epsilon ^-1)? I am further confused by Figure 1.7...
I don't understand why the subtraction of t from x, both of O(epsilon^-1), will result in a zerta of O(1)...

For those who understand these, please kindly explain them to me...

 

[mmwave]

Thank you for sharing your difficulties with understanding the concept of wave propagation and asymptotic expansion. I am a scientist who specializes in mathematical modeling and I would be happy to help you understand these concepts better.

Firstly, when it is stated that x=O(1) and t=O(1), it means that both x and t are of the same order of magnitude, which is a constant. This does not necessarily mean that x and t are bounded by a constant, but rather that they are both on the same scale. This is important because it allows us to simplify the equations and make them more manageable.

As for the domain of -infinity < x < +infinity, it is necessary to consider this entire range because we want to find a solution that works for all values of x. Even though x=O(1), it can still take on any value within that range.

The "stated condition" in page 40 refers to the condition that the function f(x) has a compact support, meaning that it is non-zero only within a finite interval. This allows us to simplify the equations and consider only values of zeta=O(1).

In page 41, when zeta=x-t=O(1), it means that the difference between x and t is on the same scale as the other variables in the equation. Therefore, if t=O(epsilon^-1), then x must also be of the same order. This is why x=O(epsilon^-1) is implied. Figure 1.7 is illustrating this concept by showing how the values of x and t change as zeta remains constant.

I hope this explanation helps to clarify your confusion. Feel free to ask any further questions or seek additional clarification. Mathematics can be challenging, but with perseverance and understanding, you will be able to comprehend it better. Keep up the good work!

 

[tacman]

Hi there,

I understand that this material can be difficult to grasp, but don't get discouraged! I will try my best to explain the concepts that are being discussed in these pages.

Firstly, when it says that x=O(1) and t=O(1), it means that both x and t are bounded by some constant value. This means that as x and t approach infinity, they will not grow without bound, but will instead be limited by some constant value. However, as you mentioned, the domain of the equation is -infinity<x<+infinity, which may seem contradictory. This is because even though x and t are bounded by a constant, they can still take on any value within that range. So while they are bounded, they are not necessarily constant themselves.

Moving on to the second point, the "stated condition" on f(x) refers to the assumption that f(x) is only non-zero for a finite range of x. This is known as a compact support, meaning that the function is only non-zero within a specific interval. As for why we only need to consider zeta=O(1), this is because we are using asymptotic expansion to solve the equation. This means that we are looking for a solution that is valid for large values of zeta, but we are only interested in the leading term as zeta approaches infinity. Therefore, we can neglect any terms that are of higher order, including those with zeta=O(epsilon). This is why we only need to consider zeta=O(1).

Finally, for the last point, when zeta=x-t=O(1) and t=O(epsilon^-1), this implies that x=O(epsilon^-1) as well. This is because t is subtracted from x, so if t is of order epsilon^-1, then x must also be of the same order in order for their difference to be of order 1. This is shown in Figure 1.7, where the two lines intersect at x=O(epsilon^-1).

I hope this helps to clarify some of your confusion. Keep at it and don't hesitate to ask for further clarification if needed. Mathematics can be challenging, but with persistence and practice, you will be able to understand it better. Best of luck!