Identifying traveling waves and their velocities in differential equations

[NutriGrainKiller]

This is for Optics - the third calc-based physics class I've taken. I'm taking calc4 now and this class seems to use differential equations quite frequently..something i am just now learning. This is an example problem that i don't quite understand:

Which of the following expressions correspond to traveling waves? For each of those, what is the speed of the wave? The quantities a, b, and c are positive constants.

a) psi(z,t) = (az - bt)^2
b) psi(x,t) = (ax + bt + c)^2
c) psi(x,t) = 1/(ax^2 + b)

now, the book attempts to explain the solution in the back of the book, but i don't quite get it. it says both a) and b) are waves since they are twice differentiable functions of (z-vt) and (x+vt) respectively. Therefore, for a) psi = a^2(z - bt/a)^2 and the velocity is b/a in the positive z-direction. For b) psi = a^2(x + bt/a + c/a)^2 and the velocity is b/a in the negative x-direction.

i don't understand where the bolded equations come from. I understand how they got the velocity and direction from those equations, but no idea how they were derived.

 

[quasar987]

You said you understand the "theory" of traveling wave, but here's a little refresher anyway...

f(x,t) describes a traveling wave if the variables x and t always appear as (x±vt) where v is a positive constant that we (rightly) call the velocity of the wave. If the variables a and t appear in f as the "combination" (x-vt), the wave is traveling in the positive x-direction. If they appear in as the combination x+vt, the wave is traveling in the negative x-direction.

Another way to see it, is this: take any function g(y) of one variable and "create" out of it the function of two variables x and t by f(x,t) = g(x±vt). Then f is traveling wave. Inversely, f(x,t) is a traveling wave if and only if there exists a function g(y) of one variable such that g(y=x±vt)=f(x,t).Ok. Now on to your problem...

a) is psi(z,t) = (az - bt)^2 a traveling wave? This boils down to asking, "Is it of the form g(z±vt)?" or equivalently, "do the variables z and t appear only in the form (z±vt)? We can see that it is by "pulling out" an 'a' out of the square. Step-by-step, this is done in the following way:

(az - bt)² =[a(z-bt/a)]²=a²(z-bt/a)²

The variables z and t appear only in the combination (z-bt/a) ==> psi(z,t) = (az - bt)^2 is a traveling wave traveling in the positive z-direction with velocity v=b/a.

Or we could reach the same conclusion by noting that if we define the function of one variable g(y)=a²y², then g(z-bt/a)=a²(z-bt/a)²=psy(z,t).Try working out b) for yourself. You just need to algebraically manipulate psi so it comes out in the form psi(x,t)=g(x±vt).

 

[quasar987]

I don'T know about you, but personally, I find the definitions of traveling wave difficult. The first one I have given lacks formalism or "rigidity". It is an attempt at conveying the idea in an intuitive way. On the other hand, the second could be too abstract for the mind to grasp fully.

In my opinion (and experience), the best way to understand what is and what is not a traveling wave is by seeing a lot of exemples:

$$f(x,t)=Ae^{-b(x-vt)^2}$$

$$f(x,t)=A\sin (b(x+vt))$$

$$f(x,t)=\frac{A\sqrt{x-vt}}{b(x-vt)^{(x-vt)}+1}$$

Are all traveling waves. However,

$$f(x,t)=Ae^{-b(x^2+vt)}$$

$$f(x,t)=A\sin(x)+ \cos(vt)$$

are not. One last exemple:

$$f(x,t)=\frac{x-vt}{x+vt}$$

is NOT a traveling wave. It has to be either x-vt OR x+vt.

 

[NutriGrainKiller]

Thanks for all the help!

After you explained it i had no trouble plowing through the rest of that problem. this was an example similar to a homework problem i was stumped on, and what you said helped me tremendously - but still can't figure out a couple parts of this homework problem.

Both contain squares inside the function (i.e...psi(x,t) and either x or t or both are squared). What are the rules for this?

also what happens if the values x and t (for psi(x,t), for example) show up more than once?

Thanks again :)

 

[quasar987]

Both contain squares inside the function (i.e...psi(x,t) and either x or t or both are squared). What are the rules for this?

also what happens if the values x and t (for psi(x,t), for example) show up more than once?

I'm not sure I know what you mean. If you'd write the exact equations you're talking about, there'd be no ambiguity.

 

[NutriGrainKiller]

I'm not sure I know what you mean. If you'd write the exact equations you're talking about, there'd be no ambiguity.

In your example https://www.physicsforums.com/latex_images/10/1067822-3.png you said it was not a traveling wave. I was wondering if it was because x was squared.

you also said that this: https://www.physicsforums.com/latex_images/10/1067822-0.png was a traveling wave. If I subtracted (b*t*x) from the end of the previous function (still exponentiated to e) would it still be a traveling wave?

the problems in question are:

psi(y,t) = e^-(a^2y^2+b^2t^2-2abty)

and

psi(z,t) = Asin(az^2-bt^2)

 

[desA]

Not all wave equations will generate two traveling waves simultaneously: f(x,t) = g(x±vt)

The number of waveforms will depend on the nature of the pde & one waveform at a time is possible.

desA

 

[HallsofIvy]

In your example
$$f(x,t)= Ae^{b(x^2+ vt)}$$ you said it was not a traveling wave. I was wondering if it was because x was squared.

More correctly because it is NOT just a function of x+vt or x-vt. There is no way to algebraically change x²+ vt into a function of x+vt alone.

you also said that this: $f(x,t)= Ae^{-b(x-vt)^2}$ was a traveling wave. If I subtracted (b*t*x) from the end of the previous function (still exponentiated to e) would it still be a traveling wave?

Do you mean $Ae^{-b(x-vt)^2- btx}$? No, it wouldn't because -b(x-vt)²- btx= -bx²+ 2bvtx- bv²t²- btx cannot be written as a function of x-vt or x+vt only.

The problems in question are:

$$psi(y,t) = e^{-(a^2y^2+b^2t^2-2abty)}$$

The crucial part is that $a^2y^2+ b^2t^2- 2abty$. Rewrite it as $a^2y^2- 2abty+ b^2t^2$ and you should be able to recognize that as a "perfect square": it is $(ay- bt)^2$. Now factor out the "a": $a^2(y- (b/a)t)^2$. Yes, that is a wave function with wave speed b/a.

and

psi(z,t) = Asin(az^2-bt^2)

az²- bt² cannot be written in terms of z-vt or z+ vt. It can be written as $a(z- \frac{b}{\sqrt{a}}t)(z+ \frac{b}{\sqrt{a}}t)$ but since those cannot be "separated"- that is psi(z,t) cannot be written as a sum of two functions, one depending only on $z- \frac{b}{\sqrt{a}}t$, the other on $z+ \frac{b}{\sqrt{a}}t$ it is not a wave function.

 

[dslowik]

Couldn't c) also be considered a wave traveling with phase velocity of 0? I think that's just semantics though..