Waves

General information about waves

There are many forms of waves, here are some examples:

1. Mechanical waves
   1. Needs a medium to propagate. Examples: - Ripples in water - Sound waves - Guitar string
2. Electromagnetic waves
   1. Does not need any medium to propagate. Examples: - Light - Radio waves - X-ray

A wave can move in two different ways:

• Transversal wave
• Longitudinal wave

Mechanical waves can be either a transversal or a longitudinal wave. Electromagnetic waves are only found as transversal waves.

Resonance

When a system experiences resonance. Something called a standing wave will occur. A standing wave, is a wave that will change with time, but will not move in space.

A standing wave can occur in one of two ways:

1. A medium is moving in the same speed, but with opposite direction of a wave.
   1. For example a wave moving in the opposite direction of a river.
2. Due to interference between two waves travelling the same speed in the opposite direction.
   1. This is the example shown in the .gif above, the black standing wave is a sum of the red and black waves.

Wave function in 1D

Derivation

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$\frac{1}{v^2}\frac{{\partial }^{2}u}{\partial t^2}=\frac{{\partial }^{2}u}{\partial x^2}$

Harmonic waves

Harmonic waves are a series of waves that are a integer multiple of the fundamental frequency, also known as the 1st harmonic.

A harmonic wave satisfies the wave equation and can be given by the expression:

$$u(x,t)=A\cos (kx-\omega t+\varphi )$$

Where;

• $A$ is the amplitude.
• $k$ is the wave number (angular change per space).
• $\omega$ is the angular frequency (angular change per time).
• $\varphi$ is the phase constant.

A harmonic wave can also be expressed in complex form. $u(x,t)=A\cos (kx-\omega t+\varphi )=\Re (A{e}^{(kx-\omega t+\varphi )i})$