Chapter 11, Problem 21Q

To determine

If the amplitude of the standing waves can be greater than the amplitude of the vibrations that cause them (up and down motion of the hand).

Answer to Problem 21Q

Solution:

Yes.

Explanation of Solution

Standing waves are those in which certain points of the wave called nodes, remain immobile. In this type of waves, the positions where the amplitude is maximum are known as antinodes, which are formed at the midpoints between two nodes.

Standing waves are the product of interference. When two waves of equal amplitude, wavelength and velocity advance in the opposite direction through a medium, standing waves form. For example, if the end of a rope is tied to a wall and the other end is waved up and down, the waves are reflected in the wall and come back in the opposite direction. If we assume that the reflection is perfectly efficient, the reflected wave will be half a wavelength delayed with respect to the initial wave. Interference will occur between both waves and the resulting displacement at any point and time will be the sum of the displacements corresponding to the incident wave and the reflected wave. At points where a crest of the incident wave coincides with a valley of the reflected one, there is no movement; These points are called nodes. Halfway between two nodes, the two waves are in phase, that is, the crests coincide with ridges and valleys with valleys; at these points, the amplitude of the resulting wave is twice as great as that of the incident wave; therefore, the string is divided by the nodes into sections of a wavelength. Between the nodes (which do not advance through the string), the string vibrates transversely.

Standing waves are formed in the strings of musical instruments that are punctured, hit or touched with a bow, as well as in the air of an organ tube and in that of a soda bottle when we blow over its mouth. Standing waves can be created in both transverse and longitudinal waves.

Conclusion:

Standing waves are the result of interference (of resonance). The phenomenon of resonance is considered the most common cause of a standing wave, where the wave energy is distributed around the antinodes and can describe large amplitude oscllations.