In am studying PDE and I have question about D'Alembert solution for one dimension wave equation.(adsbygoogle = window.adsbygoogle || []).push({});

I am going to reference Wolfram:

http://mathworld.wolfram.com/dAlembertsSolution.html

1) I want to verify the step of [tex]\frac{\partial y_0}{\partial t}[/tex] of step (14) of the page.

[tex]\Rightarrow\; \frac{\partial y_0}{\partial t}=v_0\; =\; \frac{ \partial f(x-ct)}{\partial (x-ct) } \frac{ \partial (x-ct)}{\partial t }\; + \;\frac{ \partial g(x+ct)}{\partial (x+ct) } \frac{ \partial (x+ct)}{\partial t }[/tex]

[tex] =\; c[\frac{ \partial f(x-ct)}{\partial (x-ct) }\;- \; \frac{ \partial g(x+ct)}{\partial (x+ct) }]|_{t=0} \;= \; -c\frac{ \partial f(x)}{\partial (x) }\;+ \; c\frac{ \partial g(x)}{\partial (x) } \;=\; -cf'(x)+cg'(x)\;\;\; (14)[/tex]

Am I correct on the steps?

2) I don't follow step (16)

[tex] \int_{\alpha} ^x \; v_0(s)\; ds = -cf(x) +cg(x) \;\;\; (16)[/tex]

a) Where is [tex]s[/tex] come from? Is it just a dummy variable for substitude for x and [tex]\alpha [/tex] later?

Where is [tex]\alpha[/tex] come from?? Is it supposed to be 0 instead?

b) [tex] \int_{\alpha} ^x \; v_0(s)\; ds = \int_{\alpha} ^x -cf'(x) +cg'(x) \; = [-cf(x)+cg(x)]_{\alpha}^x[/tex]

This don't agree with [tex] \int_{\alpha} ^x \; v_0(s)\; ds = -cf(x) +cg(x) \;\;\; (16)[/tex]

Please refer to (14) and (16) in Wolfram page.

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# Question regarding D'Alembert solution for one dimension wave equation

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