Longitudinal waves are the waves where the displacement of the medium is in the same direction, or in opposite direction, and direction of the travel of the wave. Mechanical waves are known as compression waves as they develop compression and refraction while travelling through any medium.

One of the main types of the wave is transverse wave where the displacement of the medium is at right angles to the direction of propagation. There are some transverse waves that are mechanical which means that these waves need a medium to travel through. These transverse mechanical waves are known as t-waves. Longitudinal waves also include sound waves and seismic P-waves. A wave along with the length of a spring is a good visualization where the distance between the coils increases or decreases.

## Sound Waves

In longitudinal harmonic sound waves, the wavelength and frequency can be described with the below-given formula,

**\(\large y(x,t)=y_{0} \cos \left ( w\left ( t-\frac{x}{c} \right ) \right )\)**

where,

y = displacement of the point on the travelling sound wave

X = distance the point travelled from the wave’s source

T = time elapsed

y0= amplitude of the oscillations

c = speed of the wave

ω = angular frequency of the wave

Quantity x/c = time ( wave takes to travel the distance x).

Frequency (f) of the wave is given by the formula,

**\(\large f=\frac{\omega }{2\pi}\)**

In Sound waves, the amplitude of the wave is the difference between the maximum pressure caused by the wave and the pressure of the undisturbed air. The propagation speed of sound depends upon the type, composition of the medium and temperature through which it propagates.

### Pressure Waves

A harmonic pressure wave oscillation will be,

**\(\large y(x,t)=y_{0} \cos \left ( kx – \omega t + \varphi \right )\)**

where,

- y0 = amplitude of displacement
- K = wavenumber
- x = distance along the axis of propagation
- ω = angular frequency
- t = time
- φ = phase difference