Is this book correct regarding waves in even dimensions?

[Roberto Valente Neto]

http://imgur.com/cUNs2z7

In this book I found by chance on Google, the author claims that “solutions of the wave equation only take the form of functions (...) in one and three dimensions. In two dimensions solutions are more complex”. Then, at the end of the paragraph of interest (which I posted the print here) he claims that in two dimensions, backward propagating terms are present but that is not the case for one and three dimensions.

As I have learned on multiple sources, the difference between odd and even dimensions is that on even dimensions the wave equation gives rise to infinite velocities for the propagation, creating a sharp wave front but diffuse tail. There’s a pretty great reddit post which clarifies exactly that (https://www.google.com.br/amp/s/amp...s/37xu2m/how_do_evendimensional_waves_behave/), but there’s no mention of backward propagating waves, just different velocities.

With that in mind, what did the author of the book mean by “backward propagating terms”? Could his interpretation be wrong? Am I missing something?

Link of the book already on the page: https://books.google.com.br/books?i...backward propagating waves dimensions&f=false

 

[Baluncore]

Welcome to PF. Sorry about the delay.
Neoclassical Physics. 2015 by Mark A. Cunningham is a good book and an interesting read. It took a while to find a copy.

When a point source on the x-axis radiates wave energy, it does so in two directions, in the positive x direction and in the negative x direction. The positive direction is conveniently referred to as “forwards”, the negative direction is then “backwards”.

The diagram on page 146, fig 5.9, demonstrates the construction of spherical wavefronts from an infinite series of wavelets. Only a forward wave propagating down the page from one side of the source A is shown. The waves, not shown, that propagate up-page, are being referred to as "backward waves".

Waves that encounter a change in propagation medium characteristics will be partially reflected. The part of the wave that continues through the medium is a continuation of the “forward wave”, the part that is reflected back towards the source is the “backward propagating term”.

 

[Roberto Valente Neto]

Baluncore, thank you for the answer. Of course, you response makes perfect sense for me. The book in question, however, seems to be implying that on even dimensions the wave naturally propagates backwards as well as forwards. It doesn’t mention reflection on medium characteristics.

On the internet, I found out that the difference between even and odd dimensions is that the wave equation, on even dimensions, gives rise to multiple velocities for the wave, making it have a sharp front (maximum velocity) but diffuse tail (other velocities till zero). That’s a sense I get from the text rather than maths because my math is relatively basic, but I don’t think I’m misunderstanding.

Read the following quote for example:
“For the case of two dimensional space this doesn't work (nor would it work with four space dimensions). We can still solve the wave equation, but the solution is not just a simple spherical wave propagating with unit velocity. Instead, we find that there are effectively infinitely many velocities, in the sense that a single pulse disturbance at the origin will propagate outward on infinitely many "light cones" (and sub-cones) with speeds ranging from the maximum down to zero. Hence if we lived in a universe with two spatial dimensions (instead of three), an observer at a fixed location from the origin of a single pulse would "see" an initial flash but then the disturbance "afterglow" would persist, becoming less and less intense, but continuing forever, as slower and slower subsidiary branches arrive.”

From this website: http://www.mathpages.com/home/kmath242/kmath242.htm

In the case this website is correct, I believe the book is wrong on this sentence. Could the author be misunderstanding or have expressed himself poorly on this case? What’s your opinion?

 

[Baluncore]

The work is an undergraduate text without much mathematics. It covers a huge field to a reasonable depth, all in one book. For that reason I think the author cannot help but express himself poorly when read by some readers. You are expected to go elsewhere for further details, which is exactly what you have done.

 

[Roberto Valente Neto]

The work is an undergraduate text without much mathematics. It covers a huge field to a reasonable depth, all in one book. For that reason I think the author cannot help but express himself poorly when read by some readers. You are expected to go elsewhere for further details, which is exactly what you have done.

Thanks. That’s kind of what I wanted to hear. Thank you so much.

 

[Baluncore]

We live in a universe of incompatible interfaces. We only see the impedance mismatches.

An author writes a book and tries to match it to a target group of readers. If the book does not match an individual's multi-dimensional state of education, the reader will dismiss the book as being too trivial or too advanced.

We feel happily “at one” with a preferred text that is well matched and convenient for us to understand. That choice of text changes with our education and understanding. We need to progressively absorb an education without reflecting too much of the information which, as must be expected, would cause standing waves.

Diverse groups of students need to be collimated and sorted prior to further education so they will fit the level of understanding required by the next study unit. At the end of each unit the student is multiplied by an examination, the degree of correlation between student and exam decides if they should propagate forwards, be reflected, or scattered. Anyone who is different will always be rejected.