Reflection/Transmission
of Wave Pulses

Here we want to visualize the reflection and the transmission of a transverse wave pulse at the boundary of two ropes of different linear densities.

Figure 7 Two ropes of different linear densities and

Suppose that the two ropes in Figure 7, having masses per unit length of and lie along the -axes in their equilibrium position and are joined at the origin, .It is easy to show that the equation describes a pulse with function traveling to the right with speed .describes a pulse traveling to the left with speed . Let us suppose that some point of a stretched rope is forced to oscillate transversely with simple harmonic motion such that a continuous succession of pulses, or a continous wave train, travels along the rope. Any transmitted or reflected wave must have the same frequency as the incident wave at the boundary of the two ropes. For the incident wave in rope 1 on the left (red one) the displacement at any time is given by

The reflected wave in rope 1 traveling to the left is given by

and the transmitted wave in rope 2 (red one) moving to the right is represented by

where and are the amplitudes of the incident, reflected and transmitted waves respectively. At the boundary, x = 0, the vertical displacement of the two ropes must be the same at every instant of time, or . This gives that

At the boundary, x = 0, the vertical forces on the ropes must be the same. If the tension F in the two ropes is the same it follows that

By substitution in the above equations, we get

,

If we define the coefficient of reflection R as the ratio of the amplitude of the reflected wave to the incident wave, then

=

The velocities in the two ropes are

and

From this it follows that

• WavePulse ( ) animates reflection and transmission of a transverse pulse at the boundary of two ropes of different linear density and

Let us illustrate what happens to a wave pulse if :

1) The mass per unit length of rope 1 is smaller than of rope 2.

> WavePulse(1,4,axes=none,scaling=unconstrained);

Figure 8 Wave pulse advancing along two ropes of different
linear densities,,

Figure 8 shows that the amplitude of the reflected pulse is opposite to that of the incident wave because of the negative value of the coefficient of reflection, R, according to the equation above.

If we let -> 0 then ->-1 and ~ - and ~ 0 as illustrated in Figure 9 where << .

> WavePulse(1,10^8,axes=none,scaling=unconstrained);

Figure 9 Wave pulse reflected at the boundary of two ropes,
linear densities ,

2) The mass per unit length of rope 1 is greater than of rope 2.

> WavePulse(4,1,axes=none,scaling=unconstrained);

Figure 10 Wave pulse advancing along two ropes with
linear densities, ,

When R is positive. There will be no change in phase at reflection, has the same sign as , as Figure 10 shows.