14.3.2. Wavelength and Angular Wave Number

The minimum distance between two points having the same phase is called the wavelength of the wave, usually denoted by λ.

Wavelength is a fundamental property of a wave, representing the spatial period of the wave – the distance over which it repeats itself. When we talk about points having the same phase, we usually refer to points where the wave reaches the same position in its cycle, such as from crest to crest or trough to trough.

You can think of a wavelength like a repeating pattern in music. Just like a song can have a repeated melody every few bars, a wave has the same repeating pattern at certain distances.

For simplicity, we can choose points of the same phase to be crests or troughs. The wavelength is then the distance between two consecutive crests or troughs in a wave.

The wavelength can be measured by observing the distance between two peaks (crests) or the distance between two valleys (troughs). This measurement provides a clear understanding of how compressed or stretched a wave is, influencing its energy and frequency.

When you're holding a long piece of string and you shake it up and down, the distance between two high points (crests) is the wavelength. This is similar to how water waves have crests and troughs when they move.

Taking φ = 0 in Eq. (14.2), the displacement at t = 0 is given by ( ,0) sin=y x a kx.

By simplifying the equation for our wave under the condition that the phase (φ) starts at zero, we can express the displacement of the wave at any position x and at time t=0. This shows the spatial behavior of the wave at a fixed moment in time.

Imagine you're at a concert looking at a wave moving through the crowd: at some moment, you can see certain parts of the crowd jumping in sync – that's like seeing the wave displacement at a fixed point in time.

Since the sine function repeats its value after every 2π change in angle, that is the displacements at points x and at 2nxkπ+ are the same, where n=1,2,3.... λ is then given by 2kπλ= or 2kπ.

The angular wavelength (k) relates directly to the physical wavelength (λ). When we express k in terms of λ, we note that the wave's structure repeats after every full cycle of 2π. This relationship allows us to convert between the wave's physical displacement properties and its angular characteristics.

Think of a Ferris wheel: every time it makes one complete revolution (2π), you've come back to where you started. Similarly, in a wave, after traveling one wavelength, it returns to the same phase.

k is the angular wave number or propagation constant; its SI unit is radian per metre or 1 radm−1.

The angular wave number (k) provides insight into how many wave cycles fit into a certain length. This helps in determining how 'tight' or 'spread out' a wave is – important in many physical applications, including optics and acoustics.

Imagine wrapping a string tightly around a cylindrical object. The tighter you wrap it, the more coils you have in a given length. Similarly, the angular wave number tells us how densely packed the cycles of the wave are.

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