# Periodic Motion

Sinusoidal waves are waves where the individual particles oscillate back and forth with a displacement that follows a sinusoidal pattern.9 They may be transverse or longitudinal. Transverse waves are those in which the direction of particle oscillation is perpendicular to the propagation (movement) of the wave. Common examples include electromagnetic waves, such as visible light, microwaves, and x-rays. A transverse wave can also be formed by attaching a string to a fixed point, and then moving your hand up and down.

In any waveform, energy is delivered in the direction of wave travel, so we can say that for a transverse wave, the particles are oscillating perpendicular to the direction of energy transfer. Longitudinal waves are ones in which the particles of the wave oscillate parallel to the direction of propagation; that is, the wave particles are oscillating in the direction of energy transfer. Sound waves are the classic example of longitudinal waves. The longitudinal wave created causes air molecules to oscillate through cycles of compression and rarefaction (decompression) along the direction of motion of the wave. A longitudinal wave can also be formed by laying a slinky flat on a table top and tapping it on one end. Waves can be described mathematically or graphically. To do so, we must first assign meaning to the physical quantities that waves represent. The distance from one maximum (crest) of the wave to the next is called the wavelength (λ). The frequency (f) is the number of wavelengths passing a fixed point per second, and is measured in hertz (Hz) or cycles per second (cps). From these two values, one can calculate the speed (ν) of a wave:

## ν = fλ

If frequency defines the number of cycles per second, then its inverse, period (T), is the number of seconds per cycle:

## T =1/f

Waves oscillate about a central point called the equilibrium position. The displacement (x) in a wave describes how far a particular point on the wave is from the equilibrium position, expressed as a vector quantity. The maximum magnitude of displacement in a wave is called its amplitude (A). The amplitude is defined as the maximum displacement from the equilibrium position to the top of a crest or bottom of a trough, not the total displacement between a crest and a trough (as this would be double the amplitude).

If we consider two waves that have the same frequency, wavelength, and amplitude and that pass through the same space at the same time, we can say that they are in phase if their respective crests and troughs coincide (line up with each other). When waves are perfectly in phase, we say that the phase difference is zero. However, if the two waves travel through the same space in such a way that the crests of one wave coincide with the troughs of the other, then we would say that they are out of phase, and the phase difference would be one-half of a wave. This could be expressed as λ/2 or, if given as an angle, 180° (one cycle = one wavelength = 360°). Waves can be out of phase with each other by any other fraction of a cycle, as well. The principle of superposition states that when waves interact with each other, the displacement of the resultant wave at any point is the sum of the displacements of the two interacting waves. When the waves are perfectly in phase, the displacements always add together and the amplitude of the resultant is equal to the sum of the amplitudes of the two waves. This is called constructive interference. When waves are perfectly out of phase, the displacements always counteract each other and the amplitude of the resultant wave is the difference between the amplitudes of the interacting waves. This is called destructive interference. If waves are not perfectly in phase or out of phase with each other, partially constructive or partially destructive interference can occur. Two waves that are nearly in phase will mostly add together. While the displacement of the resultant is simply the sum of the displacements of the two waves, the waves do not perfectly add together because they are not quite in phase. Therefore, the amplitude of the resultant wave is not quite the sum of the two waves’ amplitudes. Two waves that are almost perfectly out of phase. The two waves do not quite cancel, but the resultant wave’s amplitude is clearly much smaller than that of either of the other waves.

References

1) Kane, Thomas R.; Levinson, David A. (1996), Dynamics Online, Sunnyvale, California: Online Dynamics.    vectors

2) Resnick; Halliday; Krane (1992). Physics, Volume 1 (4th ed.). p. 83.

3) Beatty, Millard F. (2006), Principles of Engineering Mechanics, Volume 2: Dynamics—The Analysis of Motion, Mathematical Concepts and Methods in Science and Engineering, 33, Springer

4) Louis Bloomfield, Professor of Physics at the University of Virginia, How Everything Works: Making Physics Out of the Ordinary, John Wiley & Sons (2007)

5) Adkins, C.J. (1968/1983). Equilibrium Thermodynamics, third edition, McGraw-Hill, London

6) Jammer, Max (1957). Concepts of Force. Dover Publications, Inc. p. 167; footnote 14 work

7) C Hellingman (1992). “Newton’s third law revisited”. Phys. Educ. 27 (2): 112–115. Bibcode:1992 PhyEd..27.

8) Resnick & Halliday (1977). Physics (Third ed.). John Wiley & Sons. pp. 78–79. Any single force is only one aspect of a mutual interaction between two bodies.

9) Jain, Mahesh C. (2009). Textbook of Engineering Physics (Part I). p. 9. Chapter 1, p. 9

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