Wave Motion

A wave is a continuous transfer of disturbance from one part of a medium to another through successive vibrations of the particles of the medium about their mean positions. The wave carries energy and transports momentum but not the matter when it propagates in a medium.[file:1]

Mechanical Waves

The waves which require a material medium for their propagation are called mechanical waves.[file:1]

Transverse Wave

The wave in which the particles of the medium vibrate about their mean position perpendicularly to the direction of propagation of the wave is called the transverse wave. e.g. waves produced in plucked string, waves on surface of water, radio waves, etc.[file:1]

Longitudinal Wave

The wave in which the particles of the medium vibrate along the direction of propagation of the wave is called longitudinal wave. e.g. sound wave in air, wave in a spring which is suddenly compressed and released, sea waves, seismic p waves, etc.[file:1]

Progressive Wave

A progressive wave is one in which the disturbance is continuously transmitted along the direction of propagation of wave.[file:1]

Wave Properties

• Wavelength: The distance between two nearest particles of the medium vibrating in same phase is called wavelength. It is equal to the distance travelled by the wave during the time at which any particles of the medium completes one complete vibration.[file:1]
• Frequency: It is the no. of waves or crest passing through a given point per second. This is equal to no. of oscillations completed by the particles of the medium in one second.[file:1]
• Period: It is the time required for one complete wave to pass a given point. Since, f waves are produced in one second, then T = 1/f.[file:1]
• Amplitude: The maximum displacement of particles of medium from equilibrium positions when a wave passes through it is called the amplitude a of wave.[file:1]
• Wave velocity: The velocity of a wave is the distance traveled by a crest in one second. A wave travels a distance of one wavelength in a time equal to one period T. Then, wave velocity, v = λ / T = f λ.[file:1]
• Phase: It represents the current position of the wave relative to some reference position. It is measured in angles.[file:1]
• Particle velocity: The velocity of the vibrating particles of a medium when a wave is passing through it is called particle velocity V_p. Relation between particle velocity and wave velocity is V_p = -v (dy/dx). Particle velocity dy/dt changes with time but wave velocity v = f λ is constant. So, acceleration of wave is zero but that of particle is not zero.[file:1]

Equation of a Progressive Wave

Suppose a wave travelling from left to right along x-axis as shown in figure. Consider a particle at the origin O vibrating simple harmonically and its displacement at any instant t is y = a sin ωt where a is amplitude of the particle and ω its angular velocity.[file:1]

Let, another particle at P at a distance x from O. Since the disturbance will reach later to the particles to right of O, the phase of motion of these particles lags to that of O and it goes on increasing. Let, φ be the phase difference of the particle at P, then its displacement is y = a sin (ωt – φ).[file:1]

For a distance of λ, phase difference is 2π. So, the phase difference at P at a distance x from O is φ = 2π x / λ. Thus, y = a sin (ωt – 2π x / λ).[file:1]

The quantity k = 2π / λ is called the wave number or propagation constant. Then, y = a sin (ωt – kx).[file:1]

y = a sin (2π t / T – 2π x / λ) = a sin (2π (t/T – x/λ)). Again, ω = 2π f = 2π v / λ. So, y = a sin (2π v t / λ – 2π x / λ) = a sin (2π (v t – x)/λ).[file:1]

If the wave travels from right to left, equation of wave is y = a sin (2π v t / λ + 2π x / λ).[file:1]

Differential Equation of Wave Motion

The general wave motion equation is y = a sin(ωt – kx).[file:1]

Equating: ∂²y/∂t² = v² ∂²y/∂x² where v² = ω² / k² = f λ. This is the differential wave equation.[file:1]

Principle of Superposition of Waves

It states that the resultant displacement of the particle is equal to the vector sum of individual displacements due to different waves. If y be the resultant displacement of a particle and y1, y2, … are displacements due to individual waves, then y = y1 + y2 + …[file:1]

Stationary Wave

When two progressive waves of same amplitude and frequency travel in a medium in exactly the opposite direction, a resultant wave is formed. This resultant wave is called stationary wave or standing wave. The position of particle at zero displacement is called node (N) and the position of particle at which the maximum displacement takes place is called antinode (AN).[file:1]

Equation of stationary wave: Let, y1 and y2 be the displacements of two progressive waves of same amplitude a and wavelength travelling in opposite direction simultaneously with same velocity v.

Condition for Antinodes (Maximum Amplitude)

The amplitude 2a cos(kx) will be maximum if cos(kx) = 1, so kx = nπ, x = n λ / 2 (n = 0,1,2,…).[file:1]

Distance between two consecutive antinodes = λ / 2.[file:1]

Condition for Nodes (Minimum Amplitude)

The amplitude 2a cos(kx) will be minimum if cos(kx) = 0, so kx = (2n+1)π/2, x = (2n+1) λ / 4 (n = 0,1,2,…).[file:1]

Distance between two consecutive nodes = λ / 2. Distance between any consecutive node and antinode = λ / 4.[file:1]