Transverse and Longitudinal Waves: In-Depth Class 11 Notes for JEE Aspirants

Transverse waves occur when particles of the medium oscillate perpendicular to the direction of the wave's travel. 

Characteristics of Transverse Waves

• Mechanism: In a transverse wave, elements in the medium undergo a shearing strain. The layers slide past one another. That’s how we see this type of mechanical wave moving up and down vertically or sideways. The highest points of displacement from the equilibrium position are called crests, and the lowest points are called troughs.  
• Travelling Medium: Transverse waves propagate in media that can withstand shearing stress. This is a mechanical characteristic of solids with strong intermolecular forces. These waves cannot travel through fluids (liquids and gases) because fluids cannot sustain a shear force. You may want to revise the Stress Strain Curve, particularly here, which helps you visualise how materials deform under shear and other stresses. 
• Transverse Wave Examples 
  • A pulse travelling along a stretched string. The wave moves along the string's length (x-direction), but the individual segments of the string move up and down (y-direction).
  • One important and natural example of transverse waves is an S-wave (aka Secondary Waves or Shear Waves) caused by earthquakes. Such waves shake the ground perpendicular to the direction of propagation. They are only transmitted through the solid layers of the Earth. 

A longitudinal wave has particles oscillating in parallel, rather than perpendicular motion that happens with transverse waves. The medium's particles travel in the same direction as the wave travels. The movement is back and forth.   

Also, in contrast to transverse waves, longitudinal waves can travel through solids, liquids, and gases. 

Characteristics of Longitudinal Waves

• Mechanism: Longitudinal waves travel through a series of compressions and rarefactions. Compression defines a region or area where particles are pushed together, during which there is a temporary increase in density and pressure. On the other hand, we have areas like rarefaction where particles spread out and pressure along with density decreases. This change in pressure causes a restoring force that is proportional to the disturbance, and that’s how oscillations continue. This, you might recall from the principle of Simple Harmonic Motion. 
• Travelling Media: Longitudinal waves can progress through all types of elastic media that can sustain compression. That's why, we also call longitudinal waves as compression waves.

• Examples of Longitudinal Waves
  •  A vibrating source (like a piston in a pipe) compresses the air in front of it. This compression travels forward, and leaves a region of lower pressure (rarefaction) behind it.
  • To continue with another similar example as above, P-waves (primary waves) are also caused by earthquakes. Unlike S-waves which travel perpendicular to the direction of wave propagation, P-waves travel parallelly, behaving as longitudinal waves would.   

 

 

You can use this table to memorise the main differences between transverse and longitudinal waves. 

Feature	Transverse Waves	Longitudinal Waves
Particle Oscillation	Perpendicular to wave direction	Parallel to wave direction
Mechanism	Shearing Strain creating crests and troughs	Compressions & Rarefactions
Requires Shear Stress?	Yes	No
Media	Solids only (and water surface)	Solids, Liquids, and Gases
Examples	Wave on a string, Light	Sound waves, a pushed Slinky

 

 

 

Yes, some waves can be both transverse and longitudinal.

Waves on the Water Surface

These are waves on the water surface, where particles move up and down as well as back and forth.

Direction of Waves on Water Surface

The path of the waves is circular. And based on from where you view this circular path, the direction of motion is typically clockwise, towards the right. When the wave travels to the left, the direction of motion is counterclockwise.   

These waves on the surface of water are of two kinds. 

• Capillary Waves: These are ripples with short wavelengths (a few cm). The restoring force is the surface tension of the water.
• Gravity Waves: These are waves with long wavelengths (several meters). The restoring force is gravity that pulls the water back to its lowest level.

Rayleigh Waves on Solid Surface

This is also a surface wave travelling on the surface of solids. It's a type of acoustic wave travelling on the boundaries of Earth's surface and air.

Direction of Rayleigh Waves on Solid Surface

The motion follows a combination of compression and shear, characteristics of both longitudinal and transverse waves. As they are in the direction of propagation, the path is elliptical. 

 

In Class 11 Physics, you need to focus on two aspects to represent longitudinal and transverse waves. First, is the speed of these wave motions, and second, is the general equation.  

Speed of Transverse and Longitudinal Waves

One essential property of progressing waves is their speed. The properties of the medium determine that. 

In general, transverse and longitudinal waves travel at different speeds in the same medium.

Transverse Wave Speed (on Stretched String) 

A transverse wave's speed depends on the string's tension. It also depends on its mass per unit length.

Formula: v = √(T/μ)

Where:

v is the wave speed.

T is the tension in the string.

μ (mu) is the linear mass density (mass per unit length).

Longitudinal Wave's Speed (Sound Speed)

The longitudinal wave's speed depends on the medium's elastic property and density.

General Formula: v = √(B/ρ)

Where:

v is the wave speed.

B is the relevant elastic modulus of the medium.

ρ (rho) is the density of the medium.

The specific modulus B depends on the medium, i.e., fluids for longitudinal waves, while solid bars for transverse waves. 

• In fluids (liquids and gases), B is the Bulk Modulus, which relates a change in pressure to the change in volume.
• In a solid bar, B is Young's Modulus, which relates stress to longitudinal strain.

Mathematical Description of Longitudinal and Transverse Waves

Let’s see how we can mathematically describe these progressive waves. 

You will need to see them as a function in math that is dependent on position and time. 

For a simple sinusoidal wave, this function is

y(x, t) = a sin(kx - ωt + φ)

Where,

y(x, t): The displacement of a particle from its equilibrium position. For transverse waves, this is y; for longitudinal waves, it is often denoted as s(x, t).

a: The amplitude, or the maximum displacement.

k: The angular wave number.

ω (omega): The angular frequency.

φ (phi): The phase constant, which determines the wave's starting position at t=0.

This equation of transverse and longitudinal waves can also be expressed as a linear combination of sine and cosine functions:

y(x, t) = A sin(kx - ωt) + B cos(kx - ωt)

Here are some sample exercises that will strengthen your knowledge on transverse and longitudinal waves. Here you would also need to have some related knowledge on displacement relation, the wave equation, and the speed of travelling wave, to solve these straightforward ones. 

Example 1

A transverse wave on a string has velocity $20\mathrm{m}/\mathrm{s}$ and frequency 50 Hz . Calculate its wavelength. 

Solution: By using the basic wave equation, v=nλ, we get
$v=n\lambda ⟹\lambda =\frac{v}{n}=\frac{20}{50}=0.4\mathrm{m}$

 

Example 2

A string under tension 100 N with linear density $0.01\mathrm{k}\mathrm{g}/\mathrm{m}$ produces a transverse wave. Calculate the wave speed. 

Solution: Here we will use the equation for speed of a transverse wave. That is,  

$$v=\sqrt{\frac{T}{\mu }}$$

Using that, we get $v=\sqrt[]{\frac{T}{m}}=\sqrt[]{\frac{100}{0.01}}=\sqrt[]{10000}=100\mathrm{m}/\mathrm{s}$

How these Example Problems on Transverse and Longitudinal Waves Help

• In the first example, you applied the relation between wave speed, frequency, and wavelength. 
• In the second example, you connected the mechanical properties of a string, including tension and mass per unit length, to wave speed.  

Since JEE Main mainly tests students on numerical problems, you should be conceptually thorough with transverse and longitudinal waves. Additionally, we recommend revising the Oscillations Class 11 Notes, as they are foundational to understanding wave motion. 

The questions usually centre around the calculation of travelling wave speed, wavelength, and frequency for both wave types. 

For transverse waves, you will have to calculate the tension and linear density of a string. For longitudinal waves, the velocity in different media is frequently tested.

Go through Class 11 Physics Notes for quick brushups before your competitive tests. We also have some links in the next sections.