12.2 Boundary conditions  (Page 2/5)

Eventually the picture looks like the complete reflection through the $x$ -axis of what we started with:

Then all the points begin to move back. Each point on the line is oscillating up and down with a different amplitude.

If we look at the overall result, we get a standing wave.

If we superimpose the two cases where the peaks were at a maximum and the case where the same waves were at a minimum we can see thelines that the points oscillate between. We call this the envelope of the standing wave as it contains all the oscillations of the individual points.To make the concept of the envelope clearer let us draw arrows describing the motion of points along the line.

Every point in the medium containing a standing wave oscillates up and down and the amplitude of the oscillations depends on thelocation of the point. It is convenient to draw the envelope for the oscillations to describe the motion. We cannot draw the up and downarrows for every single point!

Nodes and anti-nodes

A node is a point on a wave where no displacement takes place at any time. In a standing wave, a node is a place where two waves cancel out completely as the two waves destructively interfere in thesame place. A fixed end of a rope is a node. An anti-node is a point on a wave where maximum displacement takes place. In a standingwave, an anti-node is a place where the two waves constructively interfere. Anti-nodes occur midway between nodes. A free end of a rope is an anti-node.

Node

A node is a point on a standing wave where no displacement takes place at any time. A fixed end of a rope is a node.

Anti-Node

An anti-node is a point on standing a wave where maximum displacement takes place. A free end of a rope is an anti-node.

Wavelengths of standing waves with fixed and free ends

There are many applications which make use of the properties of waves and the use of fixed and free ends. Most musical instruments rely onthe basic picture that we have presented to create specific sounds, either through standing pressure waves or standing vibratory waves instrings.

The key is to understand that a standing wave must be created in the medium that is oscillating. There are restrictions as to whatwavelengths can form standing waves in a medium.