Properties of Waves: Wave Characteristics

Waves may look similar, but what distinguishes one wave from another?

A wave is completely described using 4 characteristics.

The table below contains the 4 basic independent wave characteristics, the typical notation for each, and its definition.

Basic Characteristic	Notation	Definition
Amplitude	Depends on type of wave, we typically use A for simple mechanical waves	The maximim displacement. For example, the amplitude of a wave on a horizontal string is simply the maximum height that any part of the string attains above the string's equilibrium (rest) position (string stretched straight without any wave on it). The power that is carried by a wave is proportional to A².
Wavelength	λ	The distance, in meters, between corresponding points where the wave repeats. For example, for a wave traveling on a string, the wavelength is the distance between neighboring peaks, or neighboring troughs, or any two points where the wave returns to the same shape. Note that the distance between neighboring points on the string where the displacement is zero is NOT the wavelength. Do you see why?
Speed	v	The time rate of change of the position of a wave form, in meters/second. For example, if a wave moves horizontally along a string, the speed tells you how fast the peaks (or troughs, or any other part of the wave) are moving horizontally along the string. The speed of a wave depends on the medium in which it travels.
Phase Constant	Δφ	Shifts the wave in position and/or time. For example, two otherwise identical (same amplitude, wavelength, and speed) waves might have different values at time t=0 if their phase constants are different.

Several other important characteristics of a wave can be derived from the basic properties that are given above. These dervide quantities are listed below:

Derived

Characteristic

Notation

Definition

Period
T=λ/v.

The time it takes to produce one wavelength, or for one wavelength to pass by a fixed point in space. This depends on the wavelength and the speed of the wave. For example, a wave that has a wavlength of 6m and that is traveling at a speed 2 m/s, will take a period of 3s=(6m)/(2m/s) to pass by a fixed point in space.

Frequency

f=1/T
The number of waves produced each second,or that pass a fixed point in space, in waves/sec or Hertz (Hz). This is sometimes called the natural frequency to distinguish it from the angular frequency.

Angular Frequency

ω=2π f=2π/T
Since waves are usually periodic, one can also associate their frequency to angular rotations, with the wave repeating every 360 degrees ( 2π radians). The number of radians per second that the wave is moving/rotating is the angular frequency. Strictly speaking the units are rad/s, but since radians are dimensionless one often see the units simply written as 1/s, which is can be confusing since this is also the unit for frequency f, which is not the same!

Wavenumber

k=2π/λ
One can think of this as a spatial frequency, the number of waves contained in one meter (instead of temporal frequency which decribes the number of waves contained in one second).

Experiment with the simulation below to develop a deeper understanding of the terms we use to describe waves.

Use the sliders to change Amplitude and Wavelength. Notice what happens to the wave displayed by changing one wave characteristic at a time. Also, use the checkboxes to change the phase and get information about some of the wave characteristics. You can also "drag and drop" a wave box to highlight the wavelength and amplitude of one wave.

Use the above simulation to answer the following questions?

Derived Characteristic	Notation	Definition
Period	T=λ/v.	The time it takes to produce one wavelength, or for one wavelength to pass by a fixed point in space. This depends on the wavelength and the speed of the wave. For example, a wave that has a wavlength of 6m and that is traveling at a speed 2 m/s, will take a period of 3s=(6m)/(2m/s) to pass by a fixed point in space.
Frequency	f=1/T	The number of waves produced each second,or that pass a fixed point in space, in waves/sec or Hertz (Hz). This is sometimes called the natural frequency to distinguish it from the angular frequency.
Angular Frequency	ω=2π f=2π/T	Since waves are usually periodic, one can also associate their frequency to angular rotations, with the wave repeating every 360 degrees ( 2π radians). The number of radians per second that the wave is moving/rotating is the angular frequency. Strictly speaking the units are rad/s, but since radians are dimensionless one often see the units simply written as 1/s, which is can be confusing since this is also the unit for frequency f, which is not the same!
Wavenumber	k=2π/λ	One can think of this as a spatial frequency, the number of waves contained in one meter (instead of temporal frequency which decribes the number of waves contained in one second).