Longitudinal Waves: Definition, Formula, Types and Example

Longitudinal waves are the mechanical waves in which the constituents of the medium oscillate along the direction of wave propagation. Depending on the type of motion, the mechanical waves are classified into two types, i.e. longitudinal wave and transverse wave. The longitudinal waves are found when the energy is transmitted inside the medium whereas the transverse waves are formed at the surface.

These waves propagate in all types of media i.e. solids, liquids, and gases. Some examples of the longitudinal wave are:

• Sound wave
• The vibrations in the spring
• Earthquake’s primary waves

Key Terms:  Waves, Longitudinal waves, Transverse waves, Speed of Longitudinal waves, Particle velocity, Displacement equation, Amplitude.

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What are Longitudinal Waves?

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Longitudinal waves are waves that consist of a periodic disturbance or vibration that takes place in the same direction as the direction of the travel of the wave. The waves in which the displacement of the medium is in the same direction as the direction of the travel of the wave are known as longitudinal waves.

• Consider a long air-filled pipe with a piston at one end.
• If we suddenly move the piston rightward and then leftward, we can generate a pulse of compression and rarefaction along the pipe.
• As the motion of the elements of air is parallel to the direction of the wave’s propagation, hence wave produced is longitudinal.
• When the compression and rarefaction regions of the two waves coincide, the condition is known as constructive interference and when both regions do not coincide, the condition is known as destructive interference.

Longitudinal waves can travel in all types of media whether the media have rigidity (elasticity of shape) or not. But Transverse wave can propagate only in that medium which has rigidity i.e. tendency to oppose the change of its shape. For example:

• Inside water, there is no rigidity or tendency to oppose the change of its shape, and constituents particles of the water oscillate in the direction of the wave propagation. Hence longitudinal wave exists.
• The surface of the water has rigidity i.e. tendency to oppose the change of its shape and the constituents particle oscillate perpendicular or parallel to the direction of wave propagation. Hence it exists both longitudinal as well as transverse waves.

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Formula of Longitudinal Waves

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A longitudinal wave is a traveling wave, also referred to as a progressive wave. If during the propagation of a wave in a medium, the particles of the medium perform SHM, then the wave is called a Simple harmonic progressive wave.

If a longitudinal wave is traveling  along +x direction, then the displacement y of the element located at distance x of the medium in time t is given by

y = A sin (kx – ωt + ϕ)

And, if the longitudinal wave is traveling along the -x direction, then the displacement equation is given by

y = A sin (kx + ωt + ϕ)

Where

• y = Displacement of the particle of the medium in time t
• x = position of the particle from the wave’s source
• t = Time elapsed
• A = Maximum amplitude
• k = Propagation constant or angular wave number
• ω = Angular frequency of the wave
• ϕ = initial phase

Particle velocity and Acceleration

The velocity of the particle of the medium in a traveling longitudinal or transverse wave is called particle velocity.

For a wave equation, y = A sin (kx – ωt)

Particle velocity is given by

v_p = dy/dt = ωA cos (kx – ωt)

Where, ωA = maximum particle velocity.

Also, particle acceleration is given by

a = dv_p/dt = d²y/dt² 

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Sound Waves

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A sound wave is a type of longitudinal wave that is produced by the vibrating motion of particles traveling through a conductive medium.

• A tuning fork is an example of sound waves. 
• In Sound waves, the difference between the maximum pressure produced by the wave and the pressure of the undisturbed air is referred to as the amplitude of the wave.
• The propagation speed of sound depends upon the type of medium, the composition of the medium, and the temperature through which it is propagating.

Speed of Longitudinal wave

The speed of the longitudinal wave depends on the initial property as well as the elastic property of the medium. The longitudinal wave, such as a sound wave, travels in the form of compressions and rarefactions. Therefore, the Bulk modulus of the medium can be associated with the elastic property of the medium. And the inertial property of the medium is its mass density.

The speed of the longitudinal wave is given by

\(v= \sqrt{B \over \rho}\)

Where

• B = Bulk modulus of the medium
• ρ = mass density of the medium

According to Newton, when a sound wave propagates through a gas, the change in pressure and volume of the gas is isothermal. From Newton’s formula speed of sound in gases is given by

\(v= \sqrt{P \over \rho}\)

Where P is the pressure of the gas of the medium.

From Newton’s formula, the speed of sound in air at NTP is equal to 280 m/s.

Laplace pointed out that the pressure variation in the gases when sound wave propagates is adiabatic and not isothermal. After Laplace correction the speed of sound is given by

\(v= \sqrt{\gamma P \over \rho}\)

For air, γ = 1.40 i.e. the ratio of specific heats.

After Laplace correction, the speed of sound in air at NTP is equal to 331.3 m/s.

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Pressure Waves

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The propagation of disturbance in a medium as the pressure changes is known as a pressure wave.

The equation for a harmonic pressure wave oscillation is given as,

y(x,t) = y₀cos(kx - ωt + φ)

Where,

• y₀ = Amplitude of displacement
• k = Wavenumber
• x = Distance along the axis of propagation
• ω = Angular frequency
• t = time
• φ = Phase difference

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Characteristics of Longitudinal Waves

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Here are a few characteristics of Longitudinal Waves:

1. Compression

Compression refers to a region in a longitudinal wave where the wave’s particles are closest to each other.

2. Rarefaction

Rarefaction refers to a region in a longitudinal wave where the wave’s particles are farthest apart from each other.

3. Wavelength

The wavelength of a longitudinal wave is defined as the distance between two consecutive points. These consecutive points lie either between two compressions or between two rarefactions.

4. Amplitude

The maximum displacement of a particle from its rest point is known as its amplitude. The amplitude of a longitudinal wave is the distance from the medium’s equilibrium position to compression or rarefaction.

5. Period and Frequency

The period is defined as the amount of time taken by the wave to move one wavelength. And, the number of wavelengths per second is the frequency of the longitudinal wave.

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Examples of Longitudinal Waves

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The longitudinal waves are the type of mechanical waves that are commonly used in nature to transfer energy from one point to another in a medium. There are various examples of longitudinal waves. Some of them are as follows:

1. Sound Waves in the Air:

The sound waves are longitudinal in nature and when someone speaks the sound propagates through the medium of air and reaches the audience. The best example of longitudinal waves is the sound waves which are produced by the particles’ vibrating motion that travels through a conductive medium.

2. The Vibration in a Spring:

Suppose you have a small spring, when you will knock at the end of the spring the waves produced will start flowing through the spring. These waves will propagate inside the spring and are therefore referred to as longitudinal waves.

3. Earthquake’s Primary Waves:

It is believed that animals can sense the waves of earthquakes way before humans. The animals have the ability to sense the seismic P waves which only travel within the earth’s interior. Even humans can feel rattles and little bumps but they are mostly imperceptible to us.

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Difference Between Longitudinal Waves and Transverse Waves

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Things to Remember

• Longitudinal waves are waves in which the displacement of the medium is in the same direction as the direction of the travel of the wave. 
• The formula of a longitudinal wave is y(x,t) = y₀ cos[ω(t−x/c)].
• A sound wave is a type of longitudinal wave that is produced by the vibrating motion of particles traveling through a conductive medium.
• The propagation of disturbance in a medium as the pressure changes is known as a pressure wave.
• The equation for a harmonic pressure wave oscillation is given as, y(x,t) = y₀c
• os(kx - ωt + φ).
• The characteristics of longitudinal waves are compression, rarefaction, wavelength, amplitude, period, and frequency.
• The longitudinal waves are the type of mechanical waves that are commonly used in nature to transfer energy from one point to another in a medium.