An Introduction to Waves
What are Waves?
Intuitively a wave is a motion or a disturbance, which causes a movement of particles from place to place. The most common example of wave movement is when a pebble is tossed in water. Water waves are formed by the disturbance that takes place on the surface of the water. On the other hand sound waves are formed when a disturbance is caused by a change in air pressure. Examples of sound waves are caused by a vibrating string. Earthquake waves are waves, which are formed by disturbances that occur in the Earth’s surface.
Another description of waves is as a disturbance that moves particles through space and time. Waves travel and displace energy from one point to another. In general waves are formed when there are sustained disturbances of the particle in a medium. This medium can be either a gravitational field or a disturbance in a solid or liquid. In waves the energy vibrations are moving away from the source. The particles in the medium move, but the medium itself does not change location. This movement creates a wave pattern that begins to travel along the medium from one particle to another.
The frequency at which each particle vibrates is the same as the frequency at which the source vibrates. The period of vibration of each individual particle in the medium is equal to the vibration of the source.
All waves exhibits similar characteristics. The amplitude which gives the height of the wave; the period which defines the time for one up and down motion of the waves; and the frequency, which tells how often there is a up and down motion. The wave velocity is defined as the speed at which the disturbance moves. Wave velocity is also referred to as the speed of the wave as it moves from one location to the other. The wavelength is defined as the length of the wave.
Types of Waves
Mechanical Waves: water waves, sound waves, and the waves that are formed when a string or rope is disturbed are known as mechanical waves. The energy is carried through mediums such as water, air, the spring or rope. Mechanical waves are classified in three categories and each type disturbs the medium differently.
i.) Transverse Waves are formed when the particles of the medium vibrates perpendicular to the direction of motion. The waves move along the string, but the spring is displaced up and down.
ii.) Longitudinal waves: The wave motion is in the same direction as the disturbance. The displacement of the spring is in the same direction as the motion of the waves.
iii.) Pulse wave: Pulse waves are formed by a single bump that travels through a medium. For example a pulse wave is formed by the toss of a rope. A particular point of the rope was at rest before the pulse reaches it. It then returns to rest after the pulse passes.
Waves encounter boundaries or obstacles in their travels through the medium in which they travel. Some waves reflect back in the medium if they meet a very rigid obstacle, other waves will pass through the medium into another medium, sometimes changing direction at the boundary. Wave interference can either be constructive or destructive. The connection of waves with equal but opposite amplitude causes destructive interferences. Constructive interference occurs when the wave displacements are in the same direction. The result of constructive interference is a wave with a higher amplitude is larger than either of the two waves.
The following gives examples of wave interference.
i.) Superposition of waves: When two or more waves move through the same space they will produce a combined effect. Each wave will maintain their own integrity when they over lap and will co-exist in the same medium with themselves being permanently changed.
ii) Standing Waves. Standing waves are formed when a train of waves are generated or when a flexible material for example a rope is attached to a more rigid form such as a wall. Since the wall is too rigid to vibrate, by shaking the rope incident and reflected waves are formed. Standing waves are the result of interference. When two sets of waves of equal amplitude and wavelength pass through each other in opposite directions, the waves passes in and out of phase with each other. The waves will reflect on each other forming regions of constructive and destructive interference.
iii.) Continuous waves: Continuous waves are waves that are capable of moving from one region to another. Continuous waves will move from a region of higher speed to a region of lower speed. The amplitude of the wave will change as a result of the change in speed and also because the wave has less energy.
iv.) Traveling Waves: A traveling wave will move along the medium. Each point in the medium will vibrate regularly by the effect of the traveling waves.
The General Characteristics of Waves:
i.) Refraction: When a wave that moves in two or three dimensions comes to a region where its speed changes, its direction of travel may change. If the wave enters a region head on then the direction does not change, but if the wave enters at a different angle then the direction of travel changes. This change of direction is called refraction
ii.) Reflection: When a wave is returned after it hits a rigid object, this is referred to as reflection. The returning or reflected wave loose some of it energy on the return journey. Some of the energy travel into the medium that carried it. If the medium (example the rope) is anchored to something rigid, then the returning wave is upside down. If the rope is flipped, then a train of waves travels down the rope. The initial part of the wave reflects from the end, the rope carries both of the waves at the same time, one traveling from the end and the other from the reflecting end. When the two waves merge, the particles on the rope get a push from each wave and move according to the net force they receive.
iii.) Diffraction: When a wave meets a barrier in its path, the waves bend around the edges of the barrier forming circular waves that radiate out the waves that spread out around the barriers. The spread of the waves around the barrier is called diffraction.
The Mathematics of Waves
The Vocabulary of waves
Frequency: The frequency of a wave refers to how often the particles in the medium vibrate when a wave passes through the medium. The frequency of vibration is the number of to and fro vibrations the wave makes in a given time period. If the complete vibration occurs in one second, the frequency is one vibration per second. The unit of frequency is called the hertz (HZ). One vibration per second is 1 hertz. Higher frequencies are measured in kilohertz, (KHZ). The highest frequency is measured in gigahertz (GHZ,) in billions of hertz.
Period refers to the time which it takes for a particle on a medium to make one complete cycle. The period is measured in units of time such as seconds, hours, days or years. Frequency and period are different but related quantities. Frequency shows how often and period tells the time it takes for the wave to complete its cycle.
Period (P) = 1/ frequency. frequency = 1/ period.
The amplitude of the wave refers to the maximum amount of displacement of a particle on the medium from its rest position. The amplitude is measured from crest to rest. The crest of a wave is the point on the medium that is the maximum amount of upward movement. The trough of the wave is the point on the medium which shows the maximum downward displacement from rest.
Speed refers to how fast an object is moving and is expressed as distance per time. The speed of a wave is the distance traveled by a given point on the wave in a given period of time. The distance traveled by a wave is measured using the units meters traveled per second. The speed of periodic wave motion is related to the frequency and wave length of the waves. The wavelength is measured by how much time passes between the arrival of one crest and the other. The equation used to measure distance is also applicable to wave length: distance = speed / time. Wave speed is given as:
wave speed = wavelength × frequency
wavelength = wave speed / frequency
These relationships relate to all types of waves.
The wave Equation
: The equation speed = wavelength × frequency is called the wave equation. It gives the mathematical relationship between the speed of the wave denoted by (v) and its wavelength (Λ) and the frequency (f). Using the symbols:
V = f × Λ
The wavelength is the distance from the top of one crest to the top of the next, the unit of measure for wavelength is either meters for waves such as ocean waves; centimeters for pond waves; and nanometers for light waves. The frequency of vibration is the number of to and fro vibrations the wave makes in a given time period. If the complete vibration occurs in one second, the frequency is one vibration per second. The unit of frequency is called the hertz (HZ). One vibration per second is 1 hertz. Higher frequencies are measured in kilohertz, (KHZ). The highest frequency is measured in gigahertz (GHZ,) in billion of hertz.
Frequency = 1/ period
Period = 1/ frequency
2. The sine curve is the mathematical model used to represent the movement of waves.