Longitudinal and transverse waves

Introduction

Waves are the disturbances produced in the air that moves with or without the actual medium. It helps the matter to flow. It transports information and energy by propagation. Especially the transmission of signals through waves, which is helpful in communication. The most familiar waves are sound, water, and seismic waves. In this article, we will learn the longitudinal and transverse waves in-depth.

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Longitudinal and transverse waves

Wave motion

When many constituents oscillate in a medium, then the disturbances occur. This motion of disturbance is called wave motion.

Types of wave motion

There are two types of wave motion

• Longitudinal waves
• Transverse waves.

Transverse waves

The motion of mechanical waves involves oscillations of constituents of the medium. If the constituents of the medium oscillate perpendicular to the direction of wave propagation, it is called transverse waves.

Transverse waves travel in the form of crest and trough alternatively. There would be a change in shape elasticity. There would be no change in the density of the medium. They occur in all states of matter like solid, liquid except the gas states. Transverse waves undergo polarisation.

Speed of transverse waves

 Let us consider the waves in the string.

T be tension that provides the restoring force in the string, which is the property of the stretched string arising due to an external force.

Mu be the linear mass density is an inertial property.

M be the mass of the string.

L be the length in which the string is divided.

The formula for the wave speed in the string is calculated by the dimensional analysis.

The dimension of Mu is  [ML-1] and with tension T, it would be [MLT-2]. When we combine these dimensions, then we get the speed v which has dimension [LT-1].

Let us assume tension(T) and linear mass density () has the relevant physical quantities.

= C T/   ….(1)

In the above equation, C is the undetermined constant of dimensional analysis. Hence, C=1

The speed of transverse waves would be

= T/

This speed of the transverse waves depends mainly on the properties of the medium tension T. It does not depend on the frequency or wavelength of the wave.

Longitudinal waves

The mechanical waves oscillate between constituents of the medium. If they oscillate along the direction of waves propagation, it is called longitudinal waves. These waves propagate in the form of rarefaction and compression.

There would be changes in volume, elasticity, and pressure. They occur in all states of matter like solid, liquid, and gas. Longitudinal waves undergo polarisation.

Speed of longitudinal waves

The constituents of the medium in longitudinal waves oscillate forward and backward in the direction of propagation of the wave. As we already know, longitudinal waves propagate in the form of rarefaction and compression.

Let B be the bulk modulus. The bulk modulus is the elastic property that determines the stress under the compressional strain.

The bulk modulus is given by

B = P/ ( V/V)

Let P be the change in pressure, which produces the volumetric strain V/V.

The bulk modulus(B) has the same dimension as pressure, expressed in pascal(Pa) as per the SI units.

The tension(T) and linear mass density (p) have the relevant physical quantities.

V = C B/ 

In the above equation, C is the undetermined constant of dimensional analysis. Hence, C=1

The speed of longitudinal waves would be

V =  B/

For a linear medium like a solid bar, the lateral expansion of the bar is negligible and is considered under longitudinal strain. The elastic modulus in Young’s modulus has the same dimension as the bulk modulus.

In a solid bar, the speed of longitudinal waves is given by

V = Y/ 

Where Y is Young’s modulus of the material of the bar.

This is well known that liquids and solids have higher speeds of sound than gases.  Because the gas has less compressibility property and higher value than the bulk modulus. It compensates for their higher densities than gases.

We can estimate the speed of sound in the ideal gas approximation.

For an ideal gas, volume V, temperature T, and pressure P are related by

P V = NKBT

Where N is the number of molecules in volume (V).

KB is the Boltzmann constant

 T is the gas temperature

The isothermal change is expressed as

VP + PV = 0

Substitute B=P,

Then in an ideal gas, the speed of a longitudinal wave  is given by

V =  P/

This derivation was proposed by Newton and is known as Newton’s formula.

Examples of Longitudinal and Transverse waves

Transverse waves- Examples

The transverse waves can be found in

• Guitar string vibration
• Water ripples
• Electromagnetic waves
• Seismic waves

Longitudinal waves – Examples

There are many examples of longitudinal waves like

• Seismic waves
• Sound waves
• Spring Vibration
• Tsunami waves

Seismic waves

Animals can sense the seismic waves which cause earthquakes. Humans can experience little bumps and rattles of the wave. The P waves are the fastest wave that requires a medium to travel either in a liquid state or solid-state. The P waves in the tectonic plates move back and forth in a longitudinal manner. This results in the creation of seismic waves.

Sound waves

The sound waves can propagate through the air medium and reach the ears. The motion of the particles is disturbed by vibration.  Example: Tuning fork.

Spring vibration

The waves will be formed on the spring when we knock the one end of the spring. If one end of the spring is fixed, the waves propagate in up and down motion resulting in transverse waves.

Conclusion

Both the longitudinal and transverse waves play a major role in explaining the concept of sound. In this article, we have seen the two types of wave motion- longitudinal and transverse waves with their example in detail.