(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

(Griffiths) Suppose you send an incident wave of specified shape [itex]g_{I}(z-v_{1}t)[/itex], down string number 1. It gives rise to a reflected wave, [itex]h_{R}(z+v_{1}t)[/itex], and a transmitted wave, [itex]g_{T}(z-v_{2}t)[/itex]. By imposing the boundary conditions 9.26 and 9.27, find [itex]h_{R}[/itex] and [itex]g_{T}[/itex].

(Boundary conditions 9.26 and 9.27 are equations expressing continuity and differentiability at z = 0).

2. Relevant equations

Continuity at [itex]z = 0[/itex] implies

[tex]g_{I}(-v_{1}t) + h_{R}(v_{1}t) = g_{T}(-v_{2}t)[/tex]

Also,

[tex]\frac{\partial g_{I}}{\partial z} = -\frac{1}{v_{1}}\frac{\partial g_{I}}{\partial t}[/tex]

[tex]\frac{\partial h_{R}}{\partial z} = \frac{1}{v_{1}}\frac{\partial h_{R}}{\partial t}[/tex]

[tex]\frac{\partial g_{T}}{\partial z} = -\frac{1}{v_{2}}\frac{\partial g_{T}}{\partial t}[/tex]

So the differentiability condition

[tex]\left(\frac{\partial g_{I}}{\partial z}\right)_{z=0^{-}} + \left(\frac{\partial h_{R}}{\partial z}\right)_{z=0^{-}} = \left(\frac{\partial g_{T}}{\partial z}\right)_{z=0^{+}}[/tex]

is equivalent to

[tex]-\frac{1}{v_{1}}\frac{\partial g_{I}}{\partial t} + \frac{1}{v_{1}}\frac{\partial h_{R}}{\partial t} = -\frac{1}{v_{2}}\frac{\partial g_{T}}{\partial t}[/tex]

3. The attempt at a solution

I am just stuck with the integration of the last equation. I know I should get something like

[tex]g_{I}(-v_{1}t) - h_{R}(v_{1}t) = \frac{v_{1}}{v_{2}}g_{T}(-v_{2}t) + C[/tex]

C = constant

But I can't exactly get to this by integrating the last equation...I get velocity squares or something (which I know from analysis of sine waves, is wrong)...I think I am wrongly applying the chain rule. Can someone please help me out here? (:zzz:)

Thanks and cheers

vivek

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# Homework Help: Simple problem on waves (Math query)

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