Introduction to waves
* Waves are series of disturbances which propagate from a source to the surroundings or a destination.
* Waves are classified into three types:
• Mechanical waves
• Electromagnetic waves
• Matter waves
* Mechanical waves are governed by Newton’s laws and need a medium to propagate.
* Electromagnetic waves do not require any medium for propagation. They can travel through vacuum too.
* Matter waves are associated with moving electrons, protons, neutrons and other fundamental particles.
Transverse and longitudinal waves
* Mechanical waves can be either transverse or longitudinal in nature.
* In a transverse wave, the particles of the medium vibrate about a mean position in a direction normal to the direction of the propagation of the wave.
* In a longitudinal wave, the particles of the medium vibrate about a mean position in a direction parallel to the direction of the propagation of the wave.
* Transverse waves on the surface of still water are called capillary waves whose wavelength measures are in centimeters only.
* Waves formed on the surface of the sea are transverse waves called gravity waves whose wavelength is of the order of meters.
Displacement relation and characteristics of a progressive wave
* The equation that represents a progressive transverse wave travelling along the positive direction of the x-axis, that is, left to right is: Y(x,t) = a sin (kx –ωt +φ)
* The equation that represents a progressive transverse wave travelling along the negative direction of the x-axis, that is right to left, is:
Y(x,t) = a sin (kx + ωt +φ )
* The point of maximum positive displacement of a particle in a progressive wave is called a crest.
* The point of maximum negative displacement of a particle in a progressive wave is called a trough.
* The maximum displacement of particles from their equilibrium or mean positions as the wave passes through them is called amplitude (A).
* Wavelength (λ) is the distance between any two consecutive troughs, crests or repetitions of the shape of the wave.
* In longitudinal waves, the particles are displaced in the direction of motion of the wave.
* The displacement equation in longitudinal waves is: (x,t) = a sin (kx – ωt + φ)
Speed of a transverse wave on a stretched string
* The speed of a progressive wave is related to wavelength and frequency (v = λµ).
* As a wave travels through a medium, the particles of the medium have to oscillate or vibrate.
* Any vibrational phenomenon is associated with mass and elasticity of the particles of the medium.
* The speed of the transverse wave on a string depends upon:
• Mass per unit length of the string → µ
• The tension in the string → T
* The speed of the wave is given by the equation: v = √T/µ
Speed of a sound
* Sound waves are longitudinal waves.
* Sound waves in air consist of propagating compressions and rarefactions of small volume elements of air.
* The speed of sound in liquids and solids is higher than in air.
* Liquids and solids are less compressible than air and have a much greater bulk modulus.
* Newton’s formula to calculate the speed of sound in an ideal gas is: ν =√p/ρ
* The velocity of sound in an ideal gas, after Laplace’s correction is expressed as: ν=√ᵞ p/ρ
Principle of superposition of waves
* The addition of wave forms or pulses to find the net or resultant wave form is known as the principle of superposition.
* When waves are propagating in the same direction along a stretched string:
• Both the waves have the same angular frequency (ω), same angular wave number (k) as they are propagating along the same thing.
• They have the same amplitude (a).
• Their phases at a given x and t differ by a constant angle φ.
* By changing the angle φ we can observe different net wave forms.
That is:
• When φ is equal to zero, we find that the amplitude of the net wave form is doubled.
• When φ is equal to π the two waves are completely out of phase the amplitude of the net wave form reduces to 0 and we get a straight line.
• When the value of φ is between 0 and π the amplitude of the net wave form keeps changing from zero to ‘2a’, where ‘a’ is the amplitude of the original wave form. The net wave form shifts by :φ/2
Reflection of waves
* When a pulse of a travelling wave strikes a rigid boundary it gets reflected.
* Reflection of waves follows the laws of reflection which are similar to laws of reflection of light.
* An incident wave at a rigid boundary is represented by: Yi (x,t) = – a sin (kx – ωt)
* A reflected wave at the rigid boundary is represented by: yr(x,t) = – a sin (kx +ωt)
* A reflected wave at an open boundary is represented by: Yr(x,t) = a sin (kx+ωt)
* When a wave is incident obliquely on the boundary between two different media, we know that a part of it is reflected and a part is transmitted to the second medium.
Standing waves and normal modes
* A standing wave is represented by: y(x,t) = (2a sin kx) cos ωt
* The collection of all possible modes is called a harmonic series.
* The positions of maximum amplitude are called antinodes.
* The distance between any two consecutive nodes or antinodes is λ/2
* The standing waves on a string of length have restricted wavelength.
* The oscillation mode with the lowest frequency is called the fundamental mode or the first harmonic.
* The collection of all possible modes is called a harmonic series.
Beats
* The phenomenon of wavering of sound intensity, when two waves of nearly the same frequencies and amplitudes propagating in the same direction are superimposed on each other is called beats.
* Beat frequency ʋbeat is expressed as: ʋbeat =ʋ1-ʋ2
