Speed of Travelling Wave: Equations and Simplified Class 11 Notes

In physics, a wave is a disturbance that travels through a medium. It transfers energy from one point to another, while there is no overall displacement of the medium. 

But this is one type of behaviour of a wave where we are not considering a boundary. 

There are other factors to consider that change how a wave behaves.  This is important to know when we want to calculate the velocity of a wave. That is, we need to be clear how it behaves in a medium. Because there are two distinct categories to learn - travelling waves and standing waves.

Standing waves are understood better when you are learning reflection of waves later in this Waves chapter, which discusses how boundaries and the superposition principle are fundamental to their formation. Up to this point, we are mainly focused on calculating the speed of a simple travelling wave that just travels with its energy and momentum.

Here is a quick table of reference to keep in mind before we start exploring wave velocity calculations. 

Difference between Travelling Waves and Standing Waves

Feature	Travelling Waves	Standing Waves
Wave Motion	Moves continuously through the medium	Does not move through the medium, while it appears stationary
Energy Transfer	Energy is transferred from one point to another	No net transfer of energy across the medium
Nodes and Antinodes	Not present	Nodes (no displacement) and antinodes (maximum displacement) form
Formation	Produced by a single source	Formed by interference of two identical travelling waves moving in opposite directions
Amplitude	Same at all points (if uniform medium)	Varies from zero at nodes to maximum at antinodes

A travelling wave is a wave in which positions of both maximum and minimum amplitude travel through the medium. It is also known as a progressive wave that represents a disturbance propagating (travelling) through space and time. 

Physical Behaviour of Travelling Wave

In a travelling wave, there is no net motion of the medium itself. What travels in a wave is the disturbance pattern. The energy and wave form move through the medium while individual particles oscillate around their equilibrium positions. 

Travelling Wave Speed Formula and Equation

To understand the speed of travelling wave, a quick revision of oscillatory motion is essential. Because there, you will be able to recall the periodic and oscillatory motion along with formulaic concepts surrounding period, displacement, and phase. 

These concepts are essential in helping you calculate how fast the wave pattern is moving. Provided, you know how long each wave is (wavelength) and how many waves pass by each second (frequency). 

And since instantaneous velocity practically tells us the speed of an object at a specific time, the speed of travelling wave formula is v = fλ (velocity = frequency × wavelength). This holds for all types of medium. Consider ocean waves, sound waves from speakers, light waves, seismic waves, and ripples on water surfaces.

The travelling wave theory describes wave propagation using the wave equation. The travelling wave equation is expressed as 

y(x,t) = A sin(kx - ωt + φ), where the wave pattern moves according to f(x ± vt) format.

In physics, we measure the speed of a travelling wave to know or calculate a specific point of a constant phase as the wave moves. Also known as phase velocity, it is entirely dependent on the properties of the medium. 

Let’s break that down. 

Consider tracking a single, identifiable point on a wave (transverse wave for this example) as it moves. Here are some necessary conditions.

• This point has a constant phase in a crest or a trough, travels along with the wave.

• When we observe the wave at two different moments in time, they are separated by a small interval (∆t). 
• We will see that the entire wave pattern has shifted by a certain distance (∆x). 

The speed of the wave (v) is simply this distance divided by the time taken.

v = ∆x / ∆t

This is the physical observation of the speed of the travelling wave. Now, how do we describe it mathematically? 

The mathematical relationship of the travelling wave speed gives you further insights and clears your concepts when preparing for your annual exams and even for later fast problem-solving speed in competitive tests. 

We learned from before, while going through the Waves chapter, that the phase of a travelling wave is represented as (kx – ωt). 

Now, for a point on the wave to have a constant phase, this entire term must remain constant.

Mathematically, we show this as 

$kx-\omega t=\; constant$

As time t changes, the position x of this point must also change. This is to keep the phase constant. 

If we consider a small change in time ∆t and position ∆x:

• k(x + ∆x) – ω(t + ∆t) = kx – ωt
• k∆x – ω∆t = 0
• k∆x = ω∆t

Rearranging this for the speed of the travelling wave from the previous section, v = ∆x/∆t, we get:

$v=\frac{\omega }{k}$

Where:

• ω (Angular Frequency): ω = 2π/T = 2πf
• k (Angular Wave Number): k = 2π/λ

Again, by substituting these values, we arrive at a more widely used wave speed formula. 

• v = (2πf) / (2π/λ) = fλ
• v = fλ (Frequency × Wavelength)

This is a universal relation for all progressive or travelling waves. 

It shows that in the time it takes the travelling wave to complete one full oscillation (the period, T), the wave pattern travels a distance equal to one wavelength (λ). 

The formula for travelling wave speed, v = fλ is always true. 

But there could be some more conditions to consider. 

• The speed (v) is determined by the properties of the medium, not by the source of the wave. 
• The source determines the frequency (f), and the medium's speed then dictates the resulting wavelength (λ).

These are two key properties of a medium that determine wave speed. 

• Elastic Property: This relates to the restoring force in the medium. It's a measure of how quickly the particles of the medium return to their equilibrium position after being disturbed. (e.g., Tension, Bulk Modulus). Go read up on elastic moduli from the Mechanical Properties of Solids chapter.
• Inertial Property: This relates to the mass or density of the medium. It's a measure of the medium's resistance to acceleration. (e.g., Linear mass density, Volume density). In any case, you may want to recap in the meaning of inertia, head over to our page on Newton's First Law. 

The general relationship then is:

Speed ∝ √ (Elastic Property / Inertial Property)

 

For a transverse wave travelling along a stretched string, the properties are:

1. Elastic Property: The Tension (T) in the string.
2. Inertial Property: The Linear Mass Density (µ), which is the mass per unit length (m/L).

Here we have to use dimensional analysis. We can combine these quantities to get the dimension of speed [LT⁻¹].

1. Dimension of Tension (T) = [MLT⁻²]
2. Dimension of Linear Mass Density (µ) = [ML⁻¹]
3. The ratio T/µ has the dimension [MLT⁻²] / [ML⁻¹] = [L²T⁻²]. 

Taking the square root gives [LT⁻¹], the dimension of speed.

This leads to the formula for speed of a transverse wave on a stretched string

v = √(T / µ)

Notice that the speed depends only on the tension and density of the string, not the frequency or amplitude of the wave.

For a longitudinal wave like sound, which propagates as compressions and rarefactions, the properties are:

1. Elastic Property: The Bulk Modulus (B), which measures resistance to compression.
2. Inertial Property: The Volume Density (ρ) of the medium.

The general formula for the speed of a longitudinal wave is below. 

v = √(B / ρ)

Sound in Different Media

1. In Solids (like a long bar): The elastic property is Young's Modulus (Y). The formula becomes: v = √(Y / ρ)
2. In Liquids and Solids vs. Gases: Liquids and solids have much higher bulk moduli (B) and Young's moduli (Y) than gases, meaning they are much harder to compress. Although they are also denser (higher ρ), the increase in the modulus is far more significant. This is why sound travels much faster in solids and liquids than in gases. Additionally, brush up on the Applications of Elastic Behaviour of Materials.

The Newton-Laplace Correction for Sound in Gases

Newton's original formula v = √(P/ρ) assumed isothermal compression, predicting sound speeds ~15% too low.

Laplace corrected this by recognising that rapid acoustic compressions are adiabatic, not isothermal. Learn more about these concepts in thermodynamic processes. 

Corrected Formula: v = √(γP/ρ)

Where γ = Cp/Cv ≈ 1.4 for air. This yields v = 331.3 m/s at STP, matching experimental values.