D'Alembert's Wave. Seems Easy Or Maybe I'm Missing Something?

[mathfied]

D'Alembert's Wave. Seems Easy Or Maybe I'm Missing Something?

Hi I have this D'Alembert's question. What I've done seems easy so I wanted to clarify just in case I'm missing something, cus it seems so easy. Second part I don't understand and would appreciate a guidance.

(PART A)

D'Alembert's Solution is:
$z(x,t) = \frac{1} {2}\left[ {F(x + ct) + F(x - ct)} \right] + \frac{1} {{2c}}\int\limits_{x - ct}^{x + ct} {G(s)ds}$
of the one dimensional wave equation:
$\frac{{\partial ^2 z}} {{\partial x^2 }} = \frac{1} {{c^2 }}\frac{{\partial ^2 z}} {{\partial t^2 }},{\text{ }} - \infty < x < \infty ,{\text{ t}} \geqslant {\text{0,}}$

with "c" a constant, when the initial conditions are:
$z(x,0) = F(x),{\text{ }}z_t (x,0) = G(x),{\text{ }} - \infty < x < \infty$

Evaluate the solution if F(x)=0 for $-\infty < x < \infty$ and
$G(x) = \left\{ {\begin{array}{*{20}c} {\frac{1} {{1 + x}}} & {\left\{ {{\text{x}} \geqslant {\text{0}}} \right\}} \\ {\text{0}} & {\left\{ {{\text{x}} < {\text{0}}} \right\}} \\ \end{array} } \right.$

(PART B)

Show the values of z(x,t) for t>0 in different regions of the (x,t)- plane. Show that the value of z(x,t) is continuous for all values of x and t, gives the correct initial value of $z_t$ and $z_t \to 0$ as |x|$\to 0$

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SOLUTION TO PART A:
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GRAPH:
http://img82.imageshack.us/img82/1728/dalemberssolutionoy2.jpg $z(x,t) = \left\{ {\begin{array}{*{20}c} {\int\limits_0^{x + ct} {\frac{1} {{1 + x}}} } & {\left\{ {x + ct > 0,x - ct < 0} \right\}} & {{\text{Region I}}} \\ 0 & {\{ x + ct < 0,x - ct < 0\} } & {{\text{Region II}}} \\ {\int\limits_{x - ct}^{x + ct} {\frac{1} {{1 + x}}} } & {\{ x + ct > 0,x - ct > 0\} } & {{\text{Region III}}} \\ \end{array} } \right.$Working out region 1:
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$\begin{gathered} {\text{Region I}}:{\text{ }}\int\limits_0^{x + ct} {\frac{1} {{1 + x}}} = \left[ {\ln (1 + x)} \right]_0^{x + ct} \hfill \\ = \ln (1 + x + ct) - \ln 1 \hfill \\ = \ln \left( {\frac{{1 + x + ct}} {1}} \right) = \ln (1 + x + ct) \hfill \\ \end{gathered}$Working out region 2:
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Region 2 = 0.Working out region 3:
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$\begin{gathered} {\text{Region III: }}\int\limits_{x - ct}^{x + ct} {\frac{1} {{1 + x}}} = \left[ {\ln (1 + x)} \right]_{x - ct}^{x + ct} \hfill \\ = \ln (1 + x + ct) - \ln (1 + x - ct) \hfill \\ = \ln \left( {\frac{{1 + x + ct}} {{1 + x - ct}}} \right) \hfill \\ \end{gathered}$


FINAL SOLUTION:
-----------------------------
$z(x,t) = \left\{ {\begin{array}{*{20}c} {\ln (1 + x + ct)} & {\left\{ {x + ct > 0,x - ct < 0} \right\}} & {{\text{Region I}}} \\ 0 & {\{ x + ct < 0,x - ct < 0\} } & {{\text{Region II}}} \\ {\ln \left( {\frac{{1 + x + ct}} {{1 + x - ct}}} \right)} & {\{ x + ct > 0,x - ct > 0\} } & {{\text{Region III}}} \\ \end{array} } \right.$
Thats part a done. As far as I know that's all there is to evaluating the question. Is there anythin else I need to add?PART B: any ideas on this please?

 

[mighty2000]

Hello there,

it's great to see that you have a good understanding of D'Alembert's solution and were able to evaluate it correctly for the given initial conditions. However, there are a few things you could add to your solution to make it more complete.

Firstly, in the solution for region I, it should be specified that x + ct > 0 and x - ct < 0, as these are the conditions for this region. Similarly, for region III, it should be mentioned that x + ct > 0 and x - ct > 0.

Secondly, for part B, you can show the values of z(x,t) for t > 0 in different regions of the (x,t) plane by plotting the graph of z(x,t) vs x for different values of t. This will give you a visual representation of how the solution behaves in different regions.

To show that the value of z(x,t) is continuous for all values of x and t, you can use the definition of continuity and show that the limit of z(x,t) as x approaches any value and t approaches any value is equal to the value of z(x,t) at that point. This will prove that the solution is continuous everywhere.

Similarly, to show that the initial value of z_t is correct, you can use the given initial conditions and take the derivative of z(x,t) with respect to t, and then substitute the values of x and t to show that it matches with the given initial value of z_t.

Finally, to show that z_t approaches 0 as |x| approaches 0, you can use the definition of a limit and show that the limit of z_t as |x| approaches 0 is equal to 0.

I hope this helps you in completing part B of your question. Keep up the good work and keep questioning! Science is all about exploring and finding answers to our questions. All the best!