Wave speed wavelength and frequency

Three central wave quantities

Every periodic wave has a wavelength, frequency, and speed. Wavelength $lambda$ is the distance between repeating points in the wave, such as crest to crest or compression to compression. Frequency $f$ is the number of cycles passing a point per second. Wave speed $v$ is the speed at which the disturbance travels.

These quantities are related by

$v=flambda$

This equation applies to many types of waves: string waves, sound waves, water waves, and electromagnetic waves. The details of what sets $v$ depend on the physical system.

Period and frequency

The period $T$ is the time for one cycle. Frequency is the reciprocal:

f=rac{1}{T}

Angular frequency is

$omega=2pi f$

Wave number is

k=rac{2pi}{lambda}

The wave speed can also be written

v=rac{omega}{k}

These angular quantities are especially useful when working with sinusoidal wave equations.

Interpreting $v=flambda$

The relationship $v=flambda$ can be understood simply. If $f$ crests pass per second, and each crest is separated by distance $lambda$, then the pattern advances by $flambda$ meters each second.

In a given medium, wave speed is often set by the medium, not by the source frequency. If frequency changes while speed remains fixed, wavelength changes.

For example, on a string with fixed tension and mass density, increasing frequency decreases wavelength.

Wave speed on a string

For transverse waves on an ideal stretched string,

v=sqrt{rac{T}{mu}}

Here $T$ is string tension and $mu$ is linear mass density, mass per unit length.

Greater tension makes waves travel faster because the restoring force is stronger. Greater mass density makes waves slower because the string has more inertia.

This formula helps explain musical instruments. Tightening a string raises wave speed and changes resonant frequencies.

Sound speed

The speed of sound depends on the medium's elasticity and inertia. In a fluid, a simplified expression is

ho}}$$ where $B$ is bulk modulus and $ ho$ is density. A larger bulk modulus means the medium resists compression strongly and transmits pressure changes faster. Larger density means more inertia and slower response. In air at room temperature, sound speed is about $$vapprox 343 m/s$$ Sound generally travels faster in liquids than gases and faster in many solids than liquids. ## Light speed Electromagnetic waves in vacuum travel at $$capprox 3.00 imes 10^8 m/s$$ In materials, light travels more slowly. The index of refraction is $$n=rac{c}{v}$$ where $v$ is light speed in the material. This change in speed is responsible for refraction. ## Dispersion In some media, wave speed depends on frequency. This is called dispersion. When dispersion occurs, different frequencies travel at different speeds. A prism separates colors because different wavelengths of light travel through glass with slightly different speeds. For non-dispersive waves, all frequencies travel at the same speed, and wave shape can remain intact. For dispersive waves, wave packets can spread out. ## The big idea Wavelength, frequency, and speed are connected by $v=flambda$. Frequency is set by the source, while wave speed is usually determined by the medium. The wavelength adjusts to match both. Understanding what controls wave speed is essential for strings, sound, light, seismic waves, and all wave-based technologies.