RES.8-005 | Fall 2012 | Undergraduate

# Vibrations and Waves Problem Solving

Traveling Waves without Damping

## Problems

### Traveling Waves without Damping

#### Problem 1

There is a horizontal taut ideal string with a length of 8 meters, fixed at the left end ($$x=0$$) and attached to a massless ring free to slide on a frictionless rod on the right end at $$x=8\ m$$. The string has a mass of 0.5 kilograms and tension of 4 Newtons, assumed to be constant.

At $$t=0$$ the profiles of the transverse displacement $$y$$ and transverse velocity, $$\dfrac{\partial y}{\partial t}$$, of the string are as shown below.

On the graph below, sketch the shape of the string at $$t = 0.25$$ sec, and at $$t = 1$$ sec. Make sure you indicate the scale for $$y$$.

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This type of problem is done most easily by decomposing waveforms into superposition of progressive waves.

#### Problem 2

Voltage, current waves propagating on a transmission line undergo reflection and transmission at a junction between lines with different characteristics. If one considers two transmission lines, joined at $$x=0$$, with inductance and capacitance (per unit length) of $$L$$ and $$C$$ for $$x < 0$$, and $$L$$ and $$C/4$$ for $$x > 0$$, then the characteristic phase velocity and impedance increase by a factor of 2 across the junction at $$x=0$$. Show that this results in transmission and reflected voltage amplitudes (relative to the incoming pulse) of

\begin{eqnarray} \nonumber T &=& \frac{4}{3} \\ \nonumber R &=& \frac{1}{3} \end{eqnarray}

Also show that, in spite of the increased amplitude of transmission and the existence of a reflected pulse, energy is conserved in this system.

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Show that the sum of the power flowing in the transmitted and reflected pulses adds up, exactly, to the incident power.

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## Course Info

Fall 2012
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