Introduction:

The wave is disturbance in the material or medium, where individual parts of material may only demonstrate periodic motion, whereas waveform itself moves through material. All waves have similar characteristics, and as all forms of wave motion follow same laws and principles, knowing fundamentals of wave motion is significant in understanding sound, light, and other kinds of waves.

Let us reconsider motion of the system of N coupled masses. If we disturb the first mass from equilibrium position, individual masses slowly start to oscillate about respective equilibrium positions. That is, neither of the masses (or connecting springs) nor system as the whole moves from position. What moves instead is the wave that carries energy. It is proved by compression and stretching of springs as wave propagates. Therefore the most significant feature of wave motion is: A wave transports energy but not matter.

Another significant feature of mechanical waves is their velocity of propagation, referred to as wave velocity. It is stated as distance covered by the wave in unit time. It is different from particle velocity, i.e. velocity with which the particles of the medium vibrate back and forth about respective equilibrium positions. Furthermore, wave velocity depends on nature of the medium in which the wave propagates. The wave has features amplitude, wavelength and frequency.

Types of Waves:

One method to classify waves is on basis of direction of movement of the individual particles of medium relative to the direction that the waves travel. Classifying waves on this basis leads to three distinguished categories: transverse waves, longitudinal waves, and surface waves.

Transverse wave:

The transverse wave is the wave in which particles of medium move in the direction perpendicular to direction that the wave moves. Assume that slinky is stretched out in horizontal direction across classroom and that the pulse is introduced in slinky on left end by vibrating first coil up and down. Energy will start to be transported through slinky from left to right. As energy is transported from left to right, individual coils of medium will be displaced upwards and downwards. In this situation particles of medium move perpendicular to direction that pulse moves. This kind of wave is the transverse wave. Transverse waves are always classified by particle motion being perpendicular to wave motion.

Longitudinal wave:

The longitudinal wave is the wave in which particles of medium move in the direction parallel to direction that wave moves. Assume that slinky is stretched out in the horizontal direction across classroom and that the pulse is introduced in slinky on left end by vibrating first coil left and right. Energy will start to be transported through slinky from left to right. As energy is transported from left to right, individual coils of the medium will be displaced leftwards and rightwards. In this situation, the particles of medium move parallel to direction that pulse moves. This kind of wave is longitudinal wave. Longitudinal waves are always classified by particle motion being parallel to wave motion.

Propagation of Waves:

Take long elastic string and fix its one end to distant wall. Hold other end tightly. Move the hand up and down. A disturbance will travel along string. This disturbance is because of up and down motion of particles of string about respective mean positions. If motion of the arm (hand) is periodic, disturbance on string is the wave with sinusoidal profile. Shape of the portion of string at intervals of T /8 is shown in figure. The waveform moves to right, as shown by broad arrow. The reason that why whole string not displaced simultaneously is time lag between different parts is because of gradual transfer of disturbance between successive particles.

It is significant to differentiate between motion of waveform and motion of the particle of string. While waveform moves with the constant speed, particles of string execute SHM. To show this difference clearly, mark nine equidistant points on initial portion of string. Suppose that this string oscillates with the period T. Tie one end of this string (at mark 1) to the vertically oscillating spring-mass system. As mass m on spring moves up and down, particles at marked positions start to oscillate one after other. In time disturbance started at first particle will reach ninth particle. This signifies that in interval T / 8, the disturbance propagates from particle at mark 1 to particle at mark 2. Likewise, in next 778 interval, the disturbance travels from panicle at mark 2 to particle at mark 3 and so on. Observations are:

(i) At t = 0, all particles are at their respective mean positions.

(ii) At t = T, first, fifth and ninth particles are at their respective mean positions. First and ninth particles are about to move upward while fifth particle is about to move downward. Third and seventh particles are at position of maximum displacement but on opposite sides of horizontal axis. Positions of third and the seventh particles signify the trough and & crest, respectively. The significant point to note here is that while wave moves along string all panicles of string are oscillating up and down about respective equilibrium positions with same period (D and amplitude (a). This wave remains progressive till it reaches fixed end. We would now like you to know to represent a wave graphically as well as mathematically. This forms the subject of our discussion in the following sections.

Representation of Wave Motion:

When the wave moves along a string/spring, three parameters are involved: particle displacement, its position and time. In the 2-D graph, we can either plot displacement against time (at given position) or displacement against position (at a given time). We can easily identify that both plots are sinusoidal and have amplitude a. We can represent these as:

 y(t) = asinπ(t/T) and y(x) = asin2π(x/y)

The argument of sine function ensures that function repeats itself regularly. We can draw the analogy between wavelength and period. Wavelength is separation in space between successive in-phase points on wave. On the other hand, period is separation in time between equivalent instants in successive cycles of vibration. This signifies that wavelength and period are respectively spatial and temporal properties of the wave. It is significant to note that scales for y(x) and y(t) are different, for sound waves, displacement amplitudes are the small fraction of 1 mm while x extends to numerous metres.

In longitudinal waves, displacement of particles is along direction of wave propagation. In the figure given above the hollow circles signify mean positions of equidistant particles in medium. Arrows show their (rather magnified) longitudinal displacements at given time. Arrows are neither equal in length nor in same direction. This is clear from positions of solid circles that explain instantaneous positions of particles corresponding to heads of arrows. Displacements to right are shown in the graph towards + y axis and displacements to left towards the - y axis. For every arrow directed to the right, there is a proportionate line upward. Likewise, for every arrow directed to left, proportionate line is drawn downward. On drawing the smooth curve through heads of these lines, the graph looks like displacement-time curve for the transverse wave. At solid circles, around positions A and B, the particles have crowded together whereas around the position C, they have separated farther. These symbolize regions of compression and rarefaction. That is, there are alternate regions where density (pressure) are higher and lower than average.

Relation between Wave Velocity, Frequency and Wavelength:

The distance between successive particles vibrating in phase is called as wavelength. It is generally denoted by Greek letter λ (lambda). As wave moves a distance of one full wavelength in one period, its speed

v = wavelength/period = λ/T

AS frequency, f, is reciprocal of the period T, we can also write

v = fλ

That is, speed of any wave is equivalent to product of frequency and wavelength. This equation forecasts that in given medium, speed of the wave of given frequency is constant. This equation is derived with reference to transverse wave in a string. But it holds for all other media such as air, water, glass etc. and longitudinal waves. At STP, speeds of sound waves in air, water and steel are 332 ms^-1, 1500 ms^-1 and 5100 ms^-1, respectively. Seismic waves move with the speed of order of 6x10³ms^-1 in earth's outer crust and light moves with the speed of 3x10⁸ms^-1. That is why light which originates on or near earth reaches us almost directly.

Phase and Phase Difference:

In the periodic motion, particle displacement, velocity, and acceleration repeatedly go through cycle of changes. Different stages in the cycle may be defined in terms of phase angle. Argument of sine function is known as phase angle or simply phase. It will denote by symbol θ. Therefore, phase at x at time t in wave is provided by

θ = ω₀t - kx

Phase changes both with time and space coordinate. With time, it changes according to

Δθ = ω₀Δt = 2πfΔt for fixed x and with position according to

ΔΦ = -kΔx for fixed t

Minus sign in this equation denotes that in wave moving along +x direction, forward points lag in phase. I.e., they reach successive stages of vibration later.

Phase Velocity:

Water waves travel with constant velocity as long as properties of the medium stay constant. For harmonic progressive waves, this velocity is known as phase velocity, p v. To demonstrate this, follow the given wave crest or through as wave propagates. To keep the phase Φ (x, t) constant, look for different x as t changes. Therefore by taking differential of Φ(x, t) and setting result equal to zero, you can find relation between x and t for the point of constant phase. Differential of Φ (x, t) is given by

dΦ = ω₀dt - kdx

It will become zero provided dx and dt are related by

v_P = (dx/dt)Φ = ω₀/k

Energy Transported by Progressive Waves:

The most spectacular feature of progressive waves is that they transport energy. To compute energy carried by wave it is must to know both kinetic energy and potential energy. Instantaneous displacement of the particle is

y(x,t) = asin(ω₀t - kx).

Let us consider the thin layer of thickness and cross-sectional area A at the distance x from source. If ρ is density of the medium, the mass of the layer is ρΔxA. Thus, kinetic energy of the layer

K.E.(x,t) = 1/2mv² = 1/2ρΔxA[∂y(x,t)/∂t]²

1/2ρΔxAω₀²a²cos²(ω₀t - kx)

= 2π²f₀²ρAΔxa²cos²(ω₀t - kx)

This expression signifies that kinetic energy oscillates between zero and 1/2ρΔxAa²ω₀². This is due to value of function cos²(ω₀t - kx) differs between 0 and 1. Over one full cycle, average value of cos²θ is 1/2. So, average kinetic energy over time period is

<K.E.> = 1/4ρω₀²a²AΔx = π²a²f₀²AρΔx

To compute potential energy analytically, layer under consideration will be subject to a force

-4π²f₀²ρΔxAy(x,t)

Work done by force, when layer of interest is displaced through y from equilibrium position, is stored in the layer as its potential energy. So we can write

U(x,t) = ∫₀^yA4π²f₀²ρΔxy'dy'

=2π²f₀²ρAΔxa²sin²(ω₀t - kx)

Time-averaged potential energy of wave is

<U> = π²f₀²ρAa²Δx = <K.E.>

Total energy of wave is

E = K.E. + U

= 2π²a²f₀²ρAΔx = <K.E.> + <U>

This illustrates that half the energy of the wave is kinetic and other half is potential. If wave is moving at speed v the energy passes layer in time Δt = Δx / v. Therefore

P = 2π²a²f₀²ρvA

Intensity and Inverse Square Law:

Average rate of energy flow related with wave decreases as it spreads out. In general, it makes more sense to explain strength of wave by specifying intensity. It is stated as energy carried by the wave in unit time across a unit area normal to direction of motion.

I = 2π²a²f₀²ρv = (1/2)vV²_Aρ Where V²_A  = 2πf₀a

SI units of intensity are Jm^-2s^-1 or Wm^-2. Intensity of wave at a given position is proportional to square of amplitude at that position. For the second wave, we can write I ∝ p₀² where p₀ is maximum change in pressure over normal pressure. Area crossed by the wave increases as it spreads out. If it originates from the point source or distance from source is much greater than size of the source, area will be almost spherical (∝r²). Then principle of conservation of energy demands that E= 4πIr² be constant. So as r increases, intensity decreases as 1/r²:

I∝1/r²

p₀ ∝ 1/r

As a ∝ p₀, this relation implies that amplitude of the wave is inversely proportional to distance from the source.

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