Vibrating String of Inifinite Extent

[syj]

Homework Statement

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

In each region u_tt-c²_ju_xx=0
j=1,2
c₁≠c₂

we require continuity of u and u_x 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.

 

[mighty2000]

Thank you for your question. I understand the importance of providing examples in order to fully understand a theory. I will do my best to explain the problem and provide some examples to help you better understand the concept.

First, let's break down the problem into smaller parts. We have a string of infinite extent, which means it continues on forever in both directions (left and right). This string is made up of two homogeneous strings, which means they have the same properties (such as density) throughout. However, these two strings have different densities and are joined at x=0. This means that at x=0, there is a change in density.

Now, let's look at the equation given: utt-c2juxx=0. This is a wave equation that describes the motion of a wave along the string. The j subscript indicates that there are two regions, j=1 and j=2, which correspond to the two homogeneous strings. The c1 and c2 values are the wave speeds in each region, and since they are not equal, this means that the wave will behave differently in each region.

Next, we are told that we need continuity of u and ux at x=0. This means that the wave must be continuous (smooth) at the junction point (x=0) and its derivative (slope) must also be continuous at that point.

Now, let's move on to the initial conditions. We are given that at t<0, a wave approaches x=0 from the left. This means that at t=0, the wave is already at x=0 and is about to hit the junction point. The equation given for this is u(x,t) = F(x-c1t) when x<0 t\leq0. This is the incident wave, which is the wave that is approaching the junction point from the left.

At the junction point, the incident wave will give rise to reflected and transmitted waves. The reflected wave will bounce back towards the left and the transmitted wave will continue towards the right. At this point, the equation for the reflected wave is u(x,t) = R(x+c1t) when x<0 t>0, and for the transmitted wave, it is u(x,t) = T(x-c2t) when x>0 t>0. R and T are the reflection and transmission coefficients, respectively.

Now, for part (