Speed of a travelling wave

What is a wave?

A wave is defined as a small or large disturbance in any medium that carries energy and momentum. There is no net movement of the medium in the propagation of a wave.

A wave travelling at the maximum and minimum position of the given amplitude through a single medium is defined as a travelling wave. A travelling wave moves into space. The Maximum Speed of the noticed disturbance from the middle point, either from the top or the bottom of the trough, is defined as amplitude.

• The calculated distance between two adjacent troughs or crests in a wave is called wavelength and is written as 𝛌.
• The time period is the number of vibrations in an interval a wave takes to complete a whole vibration.
• The total number of wave vibrations in a second is defined as the frequency.
• The formula that shows the relation between frequency(f) and time period(T) is T=1/f.

The Speed of a Travelling wave

The Speed of a travelling wave is calculated in space. Take a wave moving forward in a positive direction on the x-axis. The wave equation will be y = A sin (kx – ωt) here. A will be amplitude, and k will be propagation constant. Speed is denoted by v. The propagation of waves always takes place along the positive x-axis. In a transverse wave, the wave is denoted by the y-axis. x shows the direction of the wave.

The y-axis shows the displacement of the particles. In the longitude wave, motion of particles is  along the x-axis while for transverse waves, particle’s motion is along the y axis. The equation for both transverse and longitudinal waves remains the same. When we take peaks of the phase, it does not play any role. The phase is supposed to be constant.

y/a=sin(kx – ωt) 

=> sin-1 (y/a) = (kx- ωt )

=> x = (1/k)sin-1 (y/a) +ωt/k

where (1/k)sin-1 (y/a) is constant

Speed V= dx/dt 

By differentiating with time,(1/k)sin-1 (y/a) = 0 as differentiating a constant term is 0.

V = d((1/k)sin-1 (y/a) +ωt/k)/dt

V = 0 + ω/k

V =  ω/k

Therefore, the speed of a travelling wave, V= ω/k where ω = angular frequency and k=wave number.

What is the definition of Wave velocity or Phase Velocity?

The total distance covered by the propagating wave per unit time is defined as wave velocity. To know about the travelling of waves in the direction of the positive X. y = A sin (kx – ωt) shows our travelling wave. Let’s assume that the wave of travelling has not changed its medium. Hence the form remains constant. In the waveform movement, it displaces as it moves. For any fixed point taken into consideration, we must consider a constant argument.

If we send waves in a solid object, both transverse and longitudinal waves are sent out; the result is that longitudinal waves will move faster. Longitudinal waves such as sound are transmitted through media with velocity. All sounds of air move at the same speed, irrespective of their frequency. The velocity of light in a vacuum is also independent of its frequency; however, the velocity of light depends on the dispersion in the material.

All electromagnetic waves have the same speed of 3 x 108 ms-1 in a vacuum. The Speed of electromagnetic waves is less when they travel in a medium. Light waves travel with a speed of 300,000 kilometres per second in a vacuum.

1. WaveNumber in the Speed of electromagnetic waves

k = 2π/λ

Here λ is denoted as the wavelength of the electromagnetic wave.

2. Frequency

The frequency of the wave f = ω/ 2π

Here ω is denoted as the angular frequency

3. The linear mass density or μ

Linear mass density is defined as the mass value per unit. It is used to elaborate on several characteristics of a one-dimensional objective. You can also understand the concept of linear density in three-dimensional along with one or two particular dimensions. Consider a long, thin rod of mass. Linear density fibres can be weighed in different ways. It becomes more tedious to apply if the thread fibres are crimped or are not flat and relaxed. Finally, the density is measured with a microscope. The sample particle we use to learn and understand the linear density is usually hung between two fixed points.

Variables Affecting speed of waves.

So three variables affect the speed of electromagnetic waves:

1. Pressure
2. Temperature
3. Density of medium

Pressure

Speed of wave is given by following formula

v = P 

Here is adiabatic index

P is pressure

 is density

When we look at constant temperature PV= constant

So Pm/ = constant

m represents the mass of moving particles

P = constant

Therefore pressure does not affect the speed of the sound.

Temperature

PV = RT

P M/ =RT

PM=RT

P = RT/M

 Here P = pressure

T = temperature

R = ideal gas constant

V = volume

M = mass

= density

So,

v = P

v = RTM

Hence temperature affects the speed of sound.

Density of medium

The speed of electromagnetic waves is less in humid air.

This is when water vapour in the atmosphere has either O2 or N2 molecules because most of them are already present in the atmosphere before the water molecules.

Now the mass of the molecules is

Density = mass/volume

So if mass increases, density increases, and specific volume increases, so the water density in the water molecules is lesser. This creates more space for the sound wave to move in drier air.

Examples

A sound wave of frequency 500 Hz covers a distance of 1000 m in 5 s between points x and y. Then what will be the length of waves between x and y?

Solution:

f = 500 Hz, x = 1000 m and t = 5 s

Velocity of sound v = x/t ​= 200 m/s

But v = fλ

Hence, λ = 0.4​ m

Find the medium in which the waves will travel faster:

(1) H₂

(2) N2

(3) He

(4) O2

Answer: Notice that the speed of the wave is inversely proportional to the density of the medium. Out of all the given options, the molecular mass of H₂ is the least for the equal volumes of gases. Thus, density for this medium is the least.

Therefore, the waves will travel fastest in the H₂.

Conclusion

A wave is a disturbance between the travel transferring energy without any medium without any net motion of the medium. A wave travelling of maximum and minimum amplitude through a single medium is known as a travelling wave. Speed of a wave depends on the pressure, temperature and density of the medium.