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### Key Concepts

• Terms related to waves
• Graphical representation of waves

### Introduction:

We can study the terms related to waves by observing the mechanical waves on a rope, waves on the surface of the water, and on a slinky as they are clearly visible to us. So, these mechanical waves can serve as a model to understand the characteristics of a wave.

## Transverse waves:

In the figure, we see a single disturbance is created on a rope. A single disturbance or a bump is called a wave pulse that travels through the medium.

Here we notice that the rope is disturbed in the vertical direction but the pulse travels horizontally. A wave with this type of motion is called a transverse wave.

A transverse wave is one in which the vibrations are perpendicular to the direction of the wave’s motion.

Example: Waves generated on a string.

### Terms related to a transverse wave:

A transverse wave at any instant of time t can be represented by a graph. The X-axis represents the equilibrium position of the string. The curve shows the displacements of the string at time t.

In a transverse wave when it propagates through a medium crests and troughs, or hills and valleys are generated.

Amplitude- It is the maximum displacement of the wave from its mean position.

Frequency- It is the number of vibrations in 1 second. It is denoted by f or ν. Its unit is Hertz (Hz).

Wavelength (λ) – It is the distance covered in one vibration or it is the distance between two compressions or rarefactions. Its unit is meter (m).

Time Period (T) – It is the time required to complete one vibration. Its unit is sec.

Speed of a wave = (Wavelength)/(Time Period)

Speed of a wave – V =  λ/( T)

Speed of a wave = Frequency ×Wavelength

V  = f × λ

### Propagation of longitudinal and transverse waves:

In a transverse wave when it propagates through a medium crests and troughs or hills and valleys are generated.

In a longitudinal wave when it propagates through a medium region of compressions and rarefactions are generated.

### Longitudinal waves:

In the figure, we see a single disturbance is created on a slinky by squeezing together several turns and then suddenly released. A wave pulse of closely spaced turns will move away in both directions through the medium.

Here we notice that the slinky is disturbed in the same, or parallel to the direction of the waves’ motion. A wave with this type of motion is called a longitudinal wave.

A longitudinal wave is one in which the vibrations are parallel to the direction of the wave’s motion.

Example: Waves generated on a slinky.

When a longitudinal wave travels through a slinky the coils of the slinky are tighter in some regions and looser in others.

These types of waves on a slinky (figure- a) can be represented by the curve as shown in figure (b) and are often called density waves or pressure waves.

The crests, where the spring coils are compressed. Conversely, the troughs, where the coils are stretched.

Compressions are the regions of a medium where the particles are closer together leading to high pressure and high density of particles. On the other hand, rarefactions are the regions of a medium where the particles are far apart leading to low pressure and low density of particles.

### Graphical presentation of longitudinal wave:

A longitudinal wave can also be described by a sine curve.

The graph shows the way in which the density or pressure of the medium varies when a longitudinal wave propagates through a medium.

Compressions are the regions of high pressure and density and are represented by the upper portions of the curve.

Rarefactions are the regions of low pressure and density and are represented by the lower portions of the curve, i.e., the valleys.

The peaks represent the regions of maximum compression, i.e., they represent the highest density and pressure.

The bottom-most points of the valleys represent the regions of minimum, compression or maximum rarefaction, i.e., they represent the lowest density and pressure.

### Wavelength:

The distance between any two consecutive compressions or rarefactions is called the wavelength.

The wavelength of a wave is denoted by a Greek letter “λ” called lambda.

The SI unit of measurement of wavelength is meter (m).

When a sound wave propagates through a medium the density of a medium oscillates between a maximum and a minimum value. This change in density or pressure from its maximum value to its minimum value and again to its maximum value makes one complete oscillation.

### Frequency:

The propagation of a sound wave is characterized by the motion of alternate compressions and rarefactions, i.e., crests and troughs. The number of compressions or rarefactions that cross a particular point in a unit of time is called the frequency of the wave. The frequency of a wave is denoted by a Greek letter “ν” called “nu” and is measured in the SI unit of hertz (Hz).

The parts of a wave which can be referred to as one complete oscillation are shown by a red dashed box in each picture.

Any of these can be referred to as a complete oscillation.

Time period:

The time taken by a wave to complete one oscillation is called its time period. In other words, the time taken by two consecutive compressions or rarefactions to cross a fixed point is called the time period. It is denoted by the letter T  and is measured in the SI units of seconds.

The frequency and time period are related by the following formula.

ν = 1/T

This formula can be used to calculate the frequency of a sound wave when its time period is given and vice versa.

Speed of a wave = (Wavelength)/(Time Period)

Speed of a wave – V =  λ/( T)

Speed of a wave = Frequency ×Wavelength

V  = f × λ

The speed of the wave produced in a medium remains constant, although the frequency and wavelength change. If the frequency increases the wavelength decreases.

When a pebble is dropped into a pond, the water wave that is produced carries a certain amount of energy. As the wave spreads to other parts of the pond, the energy likewise moves across the pond. Thus, the wave transfers energy from one place in the pond to another while the water remains in essentially the same place.

The rate of energy transfer depends on the amplitude of the wave. The greater the amplitude, the more energy a wave carries in a given time interval.

Energy transfer

∝∝

Square of the amplitude

Ex: when the amplitude has doubled the energy it carries in a given time interval increases by a factor of four.

Question:1

Is the amplitude of the waves increasing or decreasing along the red arrow shown?

Is the frequency of the waves increasing or decreasing along the red arrow shown?

The amplitude of the waves is decreasing along the direction shown by the red arrow. The frequency of the waves is the same along the red arrow.

Question:2

Calculate the frequency of a sound wave whose time period is 0.002 s.

Time period = 0.002 s

The frequency is given by, ν = 1/T

Therefore, frequency = 1/0.002

frequency = 2000 Hz

Question:3

Calculate the time period of the sound wave whose frequency is 100 Hz.

Frequency = 100 Hz

The frequency is given by, ν = 1/T

Thus, the time period is given by, T = 1/ν

Therefore, time period = 1/100

time period = 0.01 s

The following figure shows the relation between longitudinal and transverse waves:

### Summary

• In longitudinal waves, the medium particles vibrate parallel to the direction of propagation of the waves. In longitudinal waves, compressions, and rarefactions are generated.

• In transverse waves, the medium particles vibrate perpendicular to the direction of propagation of the waves. In transverse waves Crests (hills) and Troughs (valleys) are generated.

• Amplitude- It is the maximum displacement of the wave from its mean position.

• Frequency- It is the number of vibrations in 1 second. It is denoted by f or v. Its unit is Hertz (Hz).

• Wavelength (λ) – It is the distance covered in one vibration or it is the distance between two compressions or rarefactions. Its unit is meter (m).

• Time Period (T) – It is the time required to complete one vibration. Its unit is sec.

• Speed of a wave – V = λ/(T)=(Wavelength)/(Time Period)

• or V=f x λ=Frequency x Wavelength

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