Longitudinal waves are waves which oscillate in the direction which is parallel to the direction in which the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal waves are also called compressional or compression waves, because they produce compression and rarefaction when travelling through a medium, and pressure waves, because they produce increases and decreases in pressure. A wave along the length of a stretched Slinky toy, where the distance between coils increases and decreases, is a good visualization. Real-world examples include sound waves (vibrations in pressure, a particle of displacement, and particle velocity propagated in an elastic medium) and seismic P waves (created by earthquakes and explosions). The other main type of wave is the transverse wave, in which the displacements of the medium are at right angles to the direction of propagation. Transverse waves, for instance, describe some bulk sound waves in solid materials (but not in fluids); these are also called "shear waves" to differentiate them from the (longitudinal) pressure waves that these materials also support.
Nomenclature "Longitudinal waves" and "transverse waves" have been abbreviated by some authors as "L-waves" and "T-waves", respectively, for their own convenience. While these two abbreviations have specific meanings in seismology (L-wave for Love wave or long wave) and electrocardiography (see T wave), some authors chose to use "ℓ-waves" (lowercase 'L') and "t-waves" instead, although they are not commonly found in physics writings except for some popular science books.
Sound waves
For longitudinal harmonic sound waves, the frequency and wavelength can be described by the formula
y ( x , t ) = y o ⋅ cos ( ω ⋅ ( t − x c ) ) {\displaystyle \ y(x,t)=y_{\mathsf {o}}\cdot \cos \!{\Bigl (}\ \omega \cdot \left(t-{\tfrac {\ x\ }{c}}\right)\ {\Bigr )}\ }
where:
y {\displaystyle \ y\ ~~} is the displacement of the point on the traveling sound wave;
x {\displaystyle \ x\ ~~} is the distance from the point to the wave's source;
t {\displaystyle \ t\ ~~} is the time elapsed;
y o {\displaystyle \ y_{\mathsf {o}}\ } is the amplitude of the oscillations,
c {\displaystyle \ c\ ~~} is the speed of the wave; and
ω {\displaystyle \ \omega ~~} is the angular frequency of the wave. The quantity x c {\displaystyle \ {\frac {\ x\ }{c}}\ } is the time that the wave takes to travel the distance x . {\displaystyle \ x~.}
The ordinary frequency ( f {\displaystyle \ f\ } ) of the wave is given by
f = ω 2 π . {\displaystyle f={\frac {\omega }{\ 2\pi \ }}~.}
The wavelength can be calculated as the relation between a wave's speed and ordinary frequency.
λ = c f . {\displaystyle \lambda ={\frac {c}{\ f\ }}~.}
For sound waves, the amplitude of the wave is the difference between the pressure of the undisturbed air and the maximum pressure caused by the wave. Sound's propagation speed depends on the type, temperature, and composition of the medium through which it propagates.
Speed of longitudinal waves
Isotropic medium For isotropic solids and liquids, the speed of a longitudinal wave can be described by
v ℓ = E ℓ ρ {\displaystyle \ v_{\ell }={\sqrt {{\frac {~E_{\ell }\ }{\rho }}\ }}\ }
where
E ℓ {\displaystyle \ E_{\ell }\ ~~} is the elastic modulus, such that E ℓ = K b + 4 G 3 {\displaystyle \ E_{\ell }=K_{b}+{\frac {\ 4G\ }{3}}\ }
… excerpt ends here. Continue reading the full article.
