## Wave Motion

The easiest wave to visualize is a water wave. When a pebble is dropped in a calm pool of water, ripples travel out from the point where the pebble enters the water. The disturbance travels out from the center of the pattern, but the water does not travel with the wave. Mechanical waves—such as water waves, waves on a rope, waves in a spring, and sound waves—have two general characteristics:

• A disturbance is in some identifiable medium.
• Energy is transmitted from place to place, but the medium does not travel between two places.

For the sake of simplicity, idealized one‐dimensional waves on a rope and two‐dimensional water surface waves with no friction‐like forces provide the wave model. For ease of analysis, a pulse that is a single short wave will be used to illustrate wave characteristics that also hold true for more complex waves.

## Transverse and longitudinal waves

On the left side of Figure 1, a pulse travels on a string. As the pulse passes point P on the string, the point moves up and then back to the equilibrium position. Each segment of the rope moves only perpendicular to the motion of the wave. This type of traveling wave is called a transverse wave. Figure 1

Transverse (a) and longitudinal (b) waves.

The right side of Figure shows the pulse propagated along a stretched spring. In this case, the individual points along the medium (the spring) travel back and forth parallel to the motion of the pulse. This type of traveling wave is called a longitudinal wave. Sound waves are longitudinal waves.

## Wave characteristics

Important basic characteristics of waves are wavelength, amplitude, period, and frequency. Wavelength is the length of the repeating wave shape. Amplitude is the maximum displacement of the particles of the medium, which is determined by the energy of the wave. Figure 2 illustrates the wavelength (represented by λ the Greek letter lambda) and the amplitude (by A) for both transverse and longitudinal waves. Figure 2

Wavelength λ and amplitude A for transverse and longitudinal waves.

The period ( T) is the time for one wave to pass a given point. Period is measured in seconds. Frequency of the wave ( f) is the number of waves passing a given point in a unit of time. Frequency is measured in cycles per second or the SI unit of hertz (Hz) with the dimensions of sec −1. For example, a wave generated at 60 cycles per second has a frequency of 60 Hz and can be expressed as 60/s. Frequency is the reciprocal of the period: From the definition of velocity as distance/time (distance divided by time)—for all types of waves—the velocity is given by the following: This equation states that the wave will advance the distance of one wavelength in the time of one period of vibration. Because frequency is the reciprocal of period, velocity is also v = λ f. The velocity is dependent upon the characteristics of the medium carrying the wave.

## Superposition principle

If two waves pass through the same region of space, they combine by a process called superposition. The superposition principle is that the resultant wave formed by the simultaneous influence of two or more waves is the vector sum of the displacements due to each wave acting independently. As shown in Figure (a), if two pulses of the same size and shape on the same side of the rope arrive at a given point at the same time, they will—for an instant—combine to form a pulse that is twice the size of each of the individual pulses. This is called constructive interference. Figure 3 (b) shows what happens if the same two pulses are on opposite sides of the string. In this case, the two pulses will momentarily cancel each other out. This is called destructive interference. Figure 3

Constructive interference (a) and destructive interference (b).

## Standing waves

In Figure 4, a pulse generated by a flip of the string on the left travels to the right end, which is fixated to a wall. The pulse then reflects upside down from the fixed end. Figure 4

A wave pulse reflects from a fixed barrier.

Now, suppose that pulses are sent along the string at regular time intervals. The reflected pulse traveling to the left adds to the original pulse traveling to the right toward the wall. Standing waves are produced by the superposition of these similar but inverted pulses that are traveling in opposite directions. Figure shows successive time frames as the pulses pass through each other. Figure 5

Incident and reflected waves interfere (a) to create a standing wave (b).

Note that certain points do not move. At these points, there is always a displacement in one direction from one pulse that is canceled by an equal and opposite displacement from a reflected pulse. These points are called nodal points (N). For waves, which are pulses that alternate direction of displacement, halfway between the nodal points are segments that move up and down with a maximum displacement of twice the amplitude of the original wave. Such points are antinodes (A). The wavelength of the component waves (original and reflected) is twice the distance between adjacent nodes in the standing (resultant) wave.