# Waves

Wave is defined as the movement of any periodic motion like a spring, a pendulum, a water wave, an electric wave, a sound wave, a light wave, etc.

A wave with constant amplitude.

Any periodic wave that has amplitude varied with time, phase sinusoidally can be expressed mathematically as

R(t , θ) = R Sin (ωt + θ)
• Minimum wave height (trough) at angle 0, π, 2π, ...
F(R,t,θ) = 0 at θ = nπ
• Maximum wave height (peak or crest) at π/2, 3π/2, ...
F(R,t,θ) = R at θ = (2n+1)π/2
• Wavelength (distance between two crests) λ = 2π.
λ = 2π - A circle or a wave
2λ = 2(2π) - Two circles or two waves
kλ = k2π - Circle k or k amount of waves
• Wave Number,
k
• Velocity (or Angular Velocity),
ω = 2πf
• Time Frequency,
f = 1 / t
• Time
t = 1 / f

Wave speed is equal to the frequency times the wavelength. It can be understood as how frequently a certain distance (the wavelength in this case) is traversed.

 $f = \frac{v}{\lambda}$

Frequency is equal to speed divided by wavelength.

 $T = \frac{1}{f}$

Period is equal to the inverse of frequency.

## Variables

 λ: wavelength (m) v: wave speed (m/s) f: frequency (1/s), (Hz) T: period (s)

## Definition of terms

 Wavelength (λ): The length of one wave, or the distance from a point on one wave to the same point on the next wave. Units: meters (m). In light, λ tells us the color. Wave speed (v): the speed at which the wave pattern moves. Units: meters per second, (m/s) Frequency of oscillation (f) (or just frequency): the number of times the wave pattern repeats itself in one second. Units: seconds-1 = (1/s) = hertz (Hz) In sound, f tells us the pitch. The inverse of frequecy is the period of oscillation. Period of oscillation (T) (or just period): duration of time between one wave and the next one passing the same spot. Units: seconds (s). The inverse of the period is frequency. Use a capital, italic T and not a lowercase one, which is used for time. Amplitude (A): the maximum height of the wave measured from the average height of the wave (the wave’s center). Unit: meters (m)

Image here

The wave’s extremes, its peaks and valleys, are called antinodes. At the middle of the wave are points that do not move, called nodes.

Examples of waves: Water waves, sound waves, light waves, seismic waves, shock waves, electromagnetic waves …

## Oscillation

A wave is said to oscillate, which means to move back and forth in a regular, repeating way. This fluctuation can be between extremes of position, force, or quantity.

Different types of waves have different types of oscillations.

Longitudinal waves: Oscillation is parallel to the direction of the wave. Examples: sound waves, waves in a spring.

Transverse waves: Oscillation is perpendicular to direction of the wave. Example: light

## Interference

When waves overlap each other it is called interference. This is divided into constructive and destructive interference.

Constructive interference: the waves line up perfectly and add to each others’ strength.

Destructive interference: the two waves cancel each other out, resulting in no wave.

## Resonance

In real life, waves usually give a mishmash of constructive and destructive interference and quickly die out. However, at certain wavelengths standing waves form, resulting in resonance. These are waves that bounce back into themselves in a strengthening way, reaching maximum amplitude.

Resonance is a special case of forced vibration when the frequency of the impressed periodic force is equal to the natural frequency of the body so that it vibrates with increased amplitude, spontaneously.