Solving the Wave Equation with Method of Characteristics

[marioooo]

Hello,

My question is about method of characteristics used in solving wave equation. I've found a book on dynamics of structures, and what I cannot understand is a part when it is talked about method of characteristic. Can somebody try to read the shoert article attached below and see if everything is written correctly. Because I have some doubts about it.

Thank you

 

[Studiot]

It would help if you did two things:

Uploaded a better resolution that could be read
Uploaded it as jpg or png.

 

[marioooo]

It would help if you did two things:

Uploaded a better resolution that could be read
Uploaded it as jpg or png.

Thanks.

 

[Studiot]

I can read it now.

I assume you are happy with equations 4.59 through 4.62?

So what exactly is your uncertainty?

 

[marioooo]

OK, the problem starts when they try to explain figure 4.5. I'm not sure how he can get solution just by going along z-ct, z+ct...

 

[marioooo]

OK, I think I'm starting to get it. But another thing is bothering me. This one is more exact. How did they get reflection and transsmission coefficient for STRESS. When they are doing it for velocity, they use continuity condition, but how to get coefficient for stress. Isn't that the same equation they were supposed to use, and also get sam solution. What do they combine to get those 2 coefficient - Rs and Ts? And why they are not the same as for velocity?
Can someone please answer this question. The article is attached below.

Thank you again

 

[marioooo]

anyone? it's pretty urgent...otherwise i wouldn't be so annoying :(

 

[Studiot]

I have printed out you first scan and now your latest three.

It was not clear ( or I didn't pick up) from the first scan that the piles in question are non homogeneous so I was a bit puzzled by the questions.

However your specific question about continuity is surely answered in the conditions attached to equation 4.68.

I'm sorry, it's a long time since I have done any sonic pile testing and it's now nearly bedtime here so I will need to read throughcarfully before I can reply further, which will not be before tomorrow morning my time (GMT+1).

 

[marioooo]

Thank you for your time and answer. The thing is that i don't can see how can they get R and T for stress, because in 4.77. and 4.78. is already involved stress continuity, so don't know how did they get different R and T...

 

[Studiot]

OK

F₁ ; F₂ ; G₁ ; G₂

Are arbitrary functions which represent solutions to the wave equation. They are multiplied by coeffiecients c, p etc.

The mathematical functions described by F and G represent the shape of the wave.

In the top section of the pile only F₁ & F₂ operate.

The intial puls is described by the forward traveling wave F₁
.
F₁ is the forward traveling wave and is the only solution present in the large part of the top section.

When the wave reaches the boundary between the two sections of pile there is a phase change of 180 and some of the wave is reflected. This is a fundamental property of traveling waves.

Another way to say this is that at the boundary F₂ exists.

F₂ is the refleced wave in the top section.

Similarly in the bottom section, beyond the boundary/interface.

The interface acts as a new source for waves in the bottom section, G₁, by Huygens principle.

Equations 4.75 & 4.76 link these four quantities by coefficients as two simultaneous linear equations.

A further simplification is assumed viz G₂ = 0 ie there is no reflected wave from the bottom.

This leaves three unknowns and two equations, which cannot be directly solved.

What can be done is to put two of the unknowns in terms of the third, which has been done in equations 4.77 & 4.78.

Thus using the initial signal, F₁ as the base

The ratio of F₂ to F₁ is the ratio of the signal reflected to the signal arriving ie the reflection coefficient R

The ratio of G₁ to F₁ is the ratio of the signal transmitted to the signal arriving ie the transmission coefficient T

They have put numbers into these expressions to yield equations 4.79 & 4.80.

Hope this helps.

 

[marioooo]

Thank you. That is the part I understand. But i don't understand how did they get expressions 4.81 and 4.82, because they are not same as 4.79 and 4.80. And those two (4.79 and 4.80) are from procedure you described. But those for stress are mistry to me. For example - how to get precisely 2 ro2c2 in 4.82?

 

[Studiot]

OK but you did refer to other equations.

The stress sigma is not F or G it is given by equations 4.64 & 4.68, as they have noted.

The ratio of transmitted or reflected to incident stress make the stress transmission and reflection coefficients.

So you have to substitute into these equations to get the expressions they note, which are, of course different from the velocity coefficients.

This treatment is rather cumbersome.
You will come across alternative treatments using acoustic impedance ( Z = rho * cee) which are rather neater.

Here is one which works out all four coefficients. Note that it uses pressure rather than stress, but this is the same thing.

go well

 

[marioooo]

thanks again, you were very helpful! :)