WAVES
CHARACTERISTICS OF WAVE
The characteristics of waves are as follows:
(i) The particles of the medium traversed by a wave execute relatively small vibrations about their mean positions but the particles are not permanently displaced in the direction of propagation of the wave.
(ii) Each successive particle of the medium executes a motion quite similar to its predecessors along/perpendicular to the line of travel of the wave.
(iii) During wave motion only transfer of energy takes place but not that of a portion of the medium.
TYPES OF WAVE
Waves are mainly of three types:
(a) mechanical or elastic waves,
(b) electromagnetic waves and
(c) matter waves.
(a) Mechanical waves
Mechanical waves can be produced or propagated only in a material medium. These waves are governed by Newton’s laws of motion. For example, waves on water surface, waves on strings, sound waves etc.
Types of Mechanical waves :
Mechanical waves are of two types:
(i) Transverse wave motion, (ii) Longitudinal wave motion
(i) Transverse wave motion
- In transverse waves the particles of the medium vibrate at right angles to the direction in which the wave propagates.
- Waves on strings, surface water waves and electromagnetic waves are transverse waves.
- In electromagnetic waves (which include light waves) the disturbance that travels is not a result of vibrations of particles but it is the oscillation of electric and magnetic fields which takes place at right angles to the direction in which the wave travels.
(ii) Longitudinal wave motion
- In these types of waves, particles of the medium vibrate to and fro about their mean position along the direction of propagation of energy.
- These are also called pressure waves.
Sound waves are longitudinal mechanical waves.
b) Electromagnetic waves
- These are the waves which require no material medium for their production and propagation, i.e., they can pass through vacuum and any other material medium.
- Common examples of electromagnetic waves are visible light; ultra–violet light; radiowaves, microwaves etc.
(c) Matter waves
WAVELENGTH
WAVE VELOCITY
- Wave velocity is the time rate of propagation of wave motion in the given medium. It is different from particle velocity.
- Wave velocity depends upon the nature of the medium.
- Wave velocity (υ) = frequency (v) x wavelength (λ)
AMPLITUDE
FREQUENCY
TIME PERIOD
THE VELOCITY OF TRANSVERSE WAVES IN A STRETCHED STRING IS GIVEN BY:
THE VELOCITY OF LONGITUDINAL WAVES IN AN ELASTIC MEDIUM IS GIVEN BY:
NEWTON'S FORMULA FOR THE VELOCITY OF SOUND IN AIR
- Speed of sound in air at STP conditions, calculated on the basis of Newton’s formula is 280 ms-1.
However, the experimentally determined value is 332 ms-1. - According to Laplace, during propagation of sound waves, the change takes place under adiabatic conditions because gases are thermal insulators and compressions and refractions are alternately taking place with a high frequency.
FACTORS INFLUENCING VELOCITY OF SOUND
The velocity of sound in any gaseous medium is affected by a large number of factors like density, pressure, temperature, humidity, wind velocity etc.
(i) The velocity of sound in a gas is inversely proportional to the square root of density of the gas.
(ii) The velocity of sound is independent of the change in pressure of the gas, provided temperature remains constant.
(iii) The velocity of sound in a gas is directly proportional to the square root of its absolute temperature.
(iv) The velocity of sound in moist air is greater than the velocity of sound in dry air.
(v) If wind flows at an angle θ to the direction of propagation of sound, the velocity of sound is v + w cos θ, where w is the velocity of wind.
GENERAL EQUATION OF PROGRESSIVE WAVES
- “A progressive wave is one which travels in a given direction with constant amplitude, i.e., without attenuation.”
- As in wave motion, the displacement is a function of space as well as time, hence displacement relation is expressed as a combined function of position and time as:
y (x,t) = A sin (kx — ωt + Ф) - We may also choose a cosine function instead of a sine function. Here A, K, ω and Ф are four constant for a given wave and are known as amplitude, angular wave number, angular frequency and initial phase angle of given wave.
RELATION BETWEEN PHASE AND PATH DIFFERENCE
THE PRINCIPLE OF SUPERPOSITION OF WAVE
STANDING WAVES OR STATIONARY WAVES
PROGRESSIVE WAVES
1. The disturbance progresses onwards; it is being handed over from particle to particle. Each particle executes the same type of vibration as the preceding one, though at a different time.
2. The waves are in the form of crests and troughs, i.e., sine/cosine functions, which move on wards with a definite velocity.
3. Every particle has the same amplitude; which it attains in its own time depending upon the progress of the wave.
4. The phase of every particle varies continuously from 0 to 2π .
5. No particle remains permanently at rest. Twice during each vibration, the particles are momentarily at rest. Different particles attain this position at different times.
6. All the particles have the same maximum velocity which they attain one after another, as the wave advances.
7. There is a regular flow of energy across every plane along the direction of propagation of the wave. The average energy in a wave is half potential and half kinetic.
STATIONARY WAVES
1. The disturbance is stationary, there being no forward or backward movement of the wave. Each particle has its own vibration characteristics.
2. The waves have the appearance of a sine/cosine function, which shrinks to a straight line, twice in each vibration. It never advances.
3. Every particle has a fixed allotted amplitude. Some have zero amplitude (nodes) ,some have maximum amplitude (antinodes) always. Each particle attains this at the same given moment.
4. All the particles in one-half of the waves have a fixed phase and all the particles in the other half of the wave have the same phase in the opposite direction simultaneously.
5. There are particles which are permanently at rest (nodes) and all other particles have their own allotted maximum displacement, which they attain simultaneously. These particles are momentarily at rest twice in each vibration, all at the same time.
6. All the particles attain their individual allotted velocities depending upon their positions, simultaneously. Two particles (nodes) in one wave form have zero velocities all the time.
7. There is no flow of energy at all, across any plane. Each particle has its own allotted individual energy.
FREQUENCY OF THE STRETCHED STRING
LAW OF LENGTH
LAW OF TENSION
LAW OF MASS
BEATS
- The phenomenon of regular rise and fall in the intensity of sound, when two waves of nearly equal frequencies travelling along the same line and in the same direction superimpose each other is called beats.
- One rise and one fall in the intensity of sound constitutes one beat and the number of beats per second is called beat frequency.
- It is given as: vb = (v1-v2)
where v1 and v2 are the frequencies of the two interfering waves; v1 being greater than v2.
DOPPLER EFFECT
IMPORTANT TABLES
