Vibrating String of Inifinite Extent

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Homework Statement


Consider a string of infinite extent made up of two homogeneous strings of different densities joined at x=0

In each region utt-c2juxx=0
j=1,2
c1≠c2

we require continuity of u and ux at x=0

suppose at t<0 a wave approaches x=0 from the left.

IE: as t approaches 0 from the negative values

u(x,t) = F(x-c1t) when x<0 t[itex]\leq[/itex]0

and = 0 when x>0 t[itex]\leq[/itex]0

as t increases further, the wave reaches x=0 and gives rise to reflected and transmitted waves.

(a) formulate the appropriate initial values for u at t=0

(b) solve the initial value problem for -∞<0<∞ t>0

(c) identify the incident, reflected, and transmitted waves in your solution and determine the reflection and transmission coefficients for the junction in terms of c1 and c2. Comment on your values in the limit c1[itex]\rightarrow[/itex]c2

Homework Equations



My textbook has a small section on infinite strings but nothing about different densities.

I also need examples in order to understand the theory.

There are no examples :(

The Attempt at a Solution


No attempt.
I need help frm the bottom up.
I need examples.
I cannot understand by just looking at the theory.
If someone can please HELP.

thanks so much guys
you're my heroes.

Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 

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