Physics Study Guide/Waves

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)

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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.This happens when angle between them is 180degrees.

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.