Introduction to Waves-Equation-Sinusoidal and Speed of Waves

Detailed notes from Toppers

Equation of a Wave

General equation of a wave

• A wave is a disturbance in matter which transmits energy without any bulk movement of the medium.
• The general equation of a wave is given by: $$y = A \sin(kx - \omega t + \phi)$$ where,
• (y) represents the displacement of the wave at a point (x) and time (t).
• (A) is the amplitude of the wave
• (k ) is the wave number
• (\omega) is the angular frequency
• (\phi) is the phase constant

Traveling wave equation

• (k=2\pi/\lambda), where (\lambda) is the wavelength.
• (\omega =2\pi f), where (f) is the frequency of the wave.
• Therefore the equation becomes: $$ y(x,t)=A\sin\left(kx-\omega t+\phi\right) =A\sin2\pi\left(\frac{x}{\lambda}-ft+\frac{\phi}{2\pi}\right)$$

Standing wave equation

• A standing wave is formed due to the superposition of two waves traveling in opposite directions but having the same frequency and amplitude.
• General equation of standing wave: $$y(x,t)=2A\cos(kx)\sin(\omega t+\phi)$$

Sinusoidal Waves

Sine wave

• A sine wave is a wave in which the displacement of the particles is a sine function of time.
• The equation of a sine wave is given by $$y = A\sin(kx-\omega t+\phi)$$

Cosine wave

• A cosine wave is a wave in which the displacement of the particles is a cosine function of time.
• The equation of a cosine wave is given by $$y = A\cos(kx-\omega t+\phi)$$

Phase difference

• Phase difference is the difference in the phase of two waves with the same frequency.
• It is measured in angle degrees

Amplitude

• Amplitude is the maximum displacement of the particles from the mean position.

Wavelength

• The wavelength is the distance between two consecutive points in the wave which are in the same phase.
• It is measured in metres.

Frequency

• Frequency is the number of waves passing a point per second.
• It is measured in hertz.

Period

• Period is the time taken to complete one wavelength.
• It is measured in seconds.

Speed of Waves

Phase velocity

• The velocity of a particular phase in the wave, equal to the rate of movement of the wave pattern that results from an interference pattern.
• Phase velocity (V_p) is given by: $$ V_p=\frac{\omega}{k} =\lambda f $$

Group velocity

• Group velocity is the speed with which a group of waves or wave packets travels through a medium.
• The rate of movement of a wave packet
• Group velocity is given by: $$V_g=\frac{d\omega}{dk}$$

Energy velocity

• Energy velocity is the speed at which energy is transmitted by a wave.
• In non-dispersive media, the energy velocity is the same as the phase velocity.
• Energy velocity is given by: $$v_{en}\ =\frac{1}{\sqrt{\epsilon_0\mu_0}}$$

Relation between Wavelength, Frequency, and Speed

• Wavelength, frequency and speed of a wave are related as: $$ v = f\lambda$$
• Where, (v) is the speed of the wave (f) is the frequency of the wave (\lambda) is the wavelength of the wave

Properties of Waves

• Reflection: Reflection occurs when a wave strikes a boundary and bounces back.
• Refraction: Refraction occurs when a wave strikes a boundary between two media of different densities.
• Diffraction: Diffraction occurs when a wave passes through a small opening or obstacle.
• Interference: Interference occurs when two waves merge together to form a new wave.
• Superposition: Superposition is the combination of two or more waves to form a new wave.

Applications of Waves

• Waves have many applications. Some of the applications of waves are :

Sound waves

• Talking and hearing depend on sound waves travelling through the air
• Sound waves are used in instruments like harmonium, flute, etc.
• Ultrasonic waves are used in the production of cinema soundtracks, submarines and bats use ultrasonic waves for navigation and detecting prey.
• Medical science heavily uses sound waves in the form of ultrasound

Light waves

• Light waves are a part of the electromagnetic spectrum.
• Light waves have applications like optical fibres, photography and lasers.
• Light waves are used in remote controls and optical communication.

Water waves

• Water waves are used for transportation and recreation.
• Tides and Tsunamis are the largest ocean waves.
• The tidal energy is an example of energy generation using water waves.

Seismic waves

• Seismic waves, commonly called earthquakes, are caused by sudden movements in the earth’s crust.
• Seismometer is an instrument used to measure the magnitude of seismic waves.

Radio waves

• Radio waves are used in wireless communications, remote controls and TV.
• Radio waves are used for long-distance communication using satellites.
• These are used in radar and microwaves.

Microwaves

• Microwaves are used for cooking and heating.
• Remote sensing satellites utilise microwaves to acquire data about the Earth’s surface and atmosphere.

References:

• NCERT Physics, Class 11, Chapter 15, Waves
• NCERT Physics, Class 12, Chapter 15, Waves