`color{blue} ✍️`If you drop a little pebble in a pond of still water, the water surface gets disturbed and disturbance does not remain confined to one place, but propagates outward along a circle.

`color{blue} ✍️` If you put some cork pieces on the disturbed surface, it is seen that the cork pieces move up and down but do not move away from the centre of disturbance.

`color{blue} ✍️`This shows that the water mass does not flow outward with the circles, but rather a moving disturbance is created.

`color{blue} ✍️`Similarly, when we speak, the sound moves outward from us, without any flow of air from one part of the medium to another. The disturbances produced in air are much less obvious and only our ears or a microphone can detect them.

`color{blue} ✍️`These patterns, which move without the actual physical transfer or flow of matter as a whole, are called waves.

`color{blue} ✍️`Waves transport energy and the pattern of disturbance has information that propagate from one point to another. All our communications essentially depend on transmission of signals through waves.

`color{blue} ✍️`Speech means production of sound waves in air and hearing amounts to their detection. Often, communication involves different kinds of waves.

`color{blue} ✍️`For example, sound waves may be first converted into an electric current signal which in turn may generate an electromagnetic wave that may be transmitted by an optical cable or via a satellite. Detection of the original signal will usually involve these steps in reverse order.

`color{blue} ✍️`The most familiar type of waves such as waves on a string, water waves, sound waves, seismic waves, etc. is the so-called mechanical waves. These waves require a medium for propagation, they cannot propagate through vacuum.

`color{blue} ✍️`They involve oscillations of constituent particles and depend on the elastic properties of the medium. The electromagnetic waves that you will learn in Class XII are a different type of wave. Electromagnetic waves do not necessarily require a medium - they can travel through vacuum. Light, radiowaves, X-rays, are all electromagnetic waves. In vacuum, all electromagnetic waves have the same speed `c`, whose value is :

`color {blue} {c = 299, 792, 458 ms^-1}`

............................... (15.1)

`color{blue} ✍️`A third kind of wave is the so-called Matter waves.

`color{blue} ✍️`They are associated with constituents of matter : electrons, protons, neutrons, atoms and molecules. They arise in quantum mechanical description of nature that you will learn in your later studies.

`color{blue} ✍️`Though conceptually more abstract than mechanical or electro-magnetic waves, they have already found applications in several devices basic to modern technology; matter waves associated with electrons are employed in electron microscopes.

`color{blue} ✍️`In this chapter we will study mechanical waves, which require a material medium for their propagation.

`color{blue} ✍️`We shall illustrate this connection through simple examples.

`color{blue} ✍️`Consider a collection of springs connected to one another as shown in Fig. 15.1. If the spring at one end is pulled suddenly and released, the disturbance travels to the other end. The first spring is disturbed from its equilibrium length.

`color{blue} ✍️`Since the second spring is connected to the first, it is also stretched or compressed, and so on. The disturbance moves from one end to the other; but each spring only executes small oscillations about its equilibrium position.
As a practical example of this situation, consider a stationary train at a railway station. Different bogies of the train are coupled to each other through a spring coupling.

`color{blue} ✍️`When an engine is attached at one end, it gives a push to the bogie next to it; this push is transmitted from one bogie to another without the entire train being bodily displaced.



`color{blue} ✍️`Now let us consider the propagation of sound waves in air. As the wave passes through air, it compresses or expands a small region of air. This causes a change in the density of that region, say `δρ`, this change induces a change in pressure, `δp`, in that region.

`color{blue} ✍️`Pressure is force per unit area, so there is a restoring force proportional to the disturbance, just like in a spring. In this case, the quantity similar to extension or compression of the spring is the change in density.

`color{blue} ✍️`If a region is compressed, the molecules in that region are packed together, and they tend to move out to the adjoining region, thereby increasing the density or creating compression in the adjoining region. Consequently, the air in the first region undergoes rarefaction.

`color{blue} ✍️`If a region is comparatively rarefied the surrounding air will rush in making the rarefaction move to the adjoining region. Thus, the compression or rarefaction moves from one region to another, making the propagation of a disturbance possible in air.

`color{blue} ✍️`In solids, similar arguments can be made. In a crystalline solid, atoms or group of atoms are arranged in a periodic lattice. In these, each atom or group of atoms is in equilibrium, due to forces from the surrounding atoms. Displacing one atom, keeping the others fixed, leads to restoring forces, exactly as in a spring. So we can think of atoms in a lattice as end points, with springs between pairs of them.

`color{blue} ✍️`In the subsequent sections of this chapter we are going to discuss various characteristic properties of waves.

`color{blue} ✍️`We have seen that motion of mechanical waves involves oscillations of constituents of the medium.

`color{blue} ✍️`If the constituents of the medium oscillate perpendicular to the direction of wave propagation, we call the wave a transverse wave. If they oscillate along the direction of wave propagation, we call the wave a longitudinal wave.

`color{blue} ✍️`Fig.15.2 shows the propagation of a single pulse along a string, resulting from a single up and down jerk. If the string is very long compared to the size of the pulse, the pulse will damp out before it reaches the other end and reflection from that end may be ignored.



`color{blue} ✍️`Fig. 15.3 shows a similar situation, but this time the external agent gives a continuous periodic sinusoidal up and down jerk to one end of the string. The resulting disturbance on the string is then a sinusoidal wave. In either case the elements of the string oscillate about their equilibrium mean position as the pulse or wave passes through them.



`color{blue} ✍️`The oscillations are normal to the direction of wave motion along the string, so this is an example of transverse wave.

`color{blue} ✍️`We can look at a wave in two ways. We can fix an instant of time and picture the wave in space. This will give us the shape of the wave as a whole in space at a given instant. Another way is to fix a location i.e. fix our attention on a particular element of string and see its oscillatory motion in time.

`color{blue} ✍️`Fig. 15.4 describes the situation for longitudinal waves in the most familiar example of the propagation of sound waves. A long pipe filled with air has a piston at one end.

`color{blue} ✍️`A single sudden push forward and pull back of the piston will generate a pulse of condensations (higher density) and rarefactions (lower density) in the medium (air). If the push-pull of the piston is continuous and periodic (sinusoidal), a sinusoidal wave will be generated propagating in air along the length of the pipe. This is clearly an example of longitudinal waves.



`color{blue} ✍️`The waves considered above, transverse or longitudinal, are travelling or progressive waves since they travel from one part of the medium to another. The material medium as a whole does not move, as already noted.

`color{blue} ✍️`A stream, for example, constitutes motion of water as a whole. In a water wave, it is the disturbance that moves, not water as a whole. Likewise a wind (motion of air as a whole) should not be confused with a sound wave which is a propagation of disturbance (in pressure density) in air, without the motion of air medium as a whole.

`color{blue} ✍️`Mechanical waves are related to the elastic properties of the medium. In transverse waves, the constituents of the medium oscillate perpendicular to wave motion causing change is shape.

`color{blue} ✍️`That is, each element of the medium in subject to shearing stress. Solids and strings have shear modulus, that is they sustain shearing stress. Fluids have no shape of their own - they yield to shearing stress. This is why transverse waves are possible in solids and strings (under tension) but not in fluids.

`color{blue} ✍️`However, solids as well as fluids have bulk modulus, that is, they can sustain compressive strain.

`color{blue} ✍️`Since longitudinal waves involve compressive stress (pressure), they can be propagated through solids and fluids. Thus a steel bar possessing both bulk and sheer elastic moduli can propagate longitudinal as well as transverse waves.

`color{blue} ✍️`But air can propagate only longitudinal pressure waves (sound). When a medium such as a steel bar propagates both longitudinal and transverse waves, their speeds can be different since they arise from different elastic moduli

`color{blue} ✍️`For mathematical description of a travelling wave, we need a function of both position `x` and time `t.` Such a function at every instant should give the shape of the wave at that instant.

`color{blue} ✍️`Also at every given location, it should describe the motion of the constituent of the medium at that location. If we wish to describe a sinusoidal travelling wave (such as the one shown in Fig. 15.3) the corresponding function must also be sinusoidal.

`color{blue} ✍️`For convenience, we shall take the wave to be transverse so that if the position of the constituents of the medium is denoted by x, the displacement from the equilibrium position may be denoted by y. A sinusoidal travelling wave is then described by :

`color{blue} {y(x,t) = asin(kx - omegat + phi )}`

................................... (15.2)

`color{blue} ✍️`The term φ in the argument of sine function means equivalently that we are considering a linear combination of sine and cosine functions:

`color {blue} {y(x,t ) = A sin(kx - omega t ) + B cos(kx - omega t ) }`

...................... (15.3)

`color{blue} ✍️`From Equations (15.2) and (15.3),

`color{blue} { a = sqrt(A^2 + B^2 ) " and " phi = tan^-1 (B/A)}`

`color{blue} ✍️`To understand why Equation (15.2) represents a sinusoidal travelling wave, take a fixed instant, say `t = t_0`. Then the argument of the sine function in Equation (15.2) is simply `kx +` constant.

`color{blue} ✍️`Thus the shape of the wave (at any fixed instant) as a function of x is a sine wave. Similarly, take a fixed location, say `x = x_0`. Then the argument of the sine function in Equation (15.2) is constant `-ωt`. The displacement y, at a fixed location, thus varies sinusoidally with time.

`color{blue} ✍️`That is, the constituents of the medium at different positions execute simple harmonic motion. Finally, as t increases, x must increase in the positive direction to keep `kx – ωt + φ` constant.

`color{blue} ✍️`Thus Eq. (15.2) represents a sinusiodal (harmonic) wave travelling along the positive direction of the x-axis. On the other hand, a function

`color{blue} {y(x,t) = a sin(kx + omega t + phi)}`

...................(15.4)

`color{blue} ✍️`represents a wave travelling in the negative direction of x-axis. Fig. (15.5) gives the names of the various physical quantities appearing in Eq. (15.2) that we now interpret.



`color{blue} ✍️`Fig. 15.6 shows the plots of Eq. (15.2) for different values of time differing by equal intervals of time. In a wave, the crest is the point of maximum positive displacement, the trough is the point of maximum negative displacement.

`color{blue} ✍️`To see how a wave travels, we can fix attention on a crest and see how it progresses with time. In the figure, this is shown by a cross (×) on the crest. In the same manner, we can see the motion of a particular constituent of the medium at a fixed location, say at the origin of the x-axis.

`color{blue} ✍️`This is shown by a solid dot `(•)` The plots of Fig. 15.6 show that with time, the solid dot (•) at the origin moves periodically i.e. the particle at the origin oscillates about its mean position as the wave progresses.

`color{blue} ✍️`This is true for any other location also. We also see that during the time the solid dot (•) has completed one full oscillation, the crest has moved further by a certain distance. Using the plots of Fig. 15.6, we now define the various quantities of Eq. (15.2).



`color {brown} bbul" Amplitude and Phase "`
`color{blue} ✍️`In Eq. (15.2), since the sine function varies between `1` and `–1,` the displacement `y (x,t)` varies between `a` and `–a`. We can take a to be a positive constant, without any loss of generality.

`color{blue} ✍️`Then a represents the maximum displacement of the constituents of the medium from their equilibrium position. Note that the displacement y may be positive or negative, but a is positive. It is called the amplitude of the wave.

`color{blue} ✍️`The quantity `(kx – ωt + φ)` appearing as the argument of the sine function in Eq. (15.2) is called the `"phase of the wave."`

`color{blue} ✍️`Given the amplitude a, the phase determines the displacement of the wave at any position and at any instant. Clearly `φ` is the phase at `x = 0` and `t = 0`. Hence `φ` is called the initial phase angle. By suitable choice of origin on the x-axis and the intial time, it is possible to have φ = 0. Thus there is no loss of generality in dropping φ, i.e., in taking Eq. (15.2) with `φ = 0`.

`color {brown} bbul" Wavelength and Angular Wave Number"`
`color{blue} ✍️`The minimum distance between two points having the same phase is called the wave length of the wave, usually denoted by `λ.`

`color{blue} ✍️`For simplicity, we can choose points of the same phase to be crests or troughs. The wavelength is then the distance between two consecutive crests or troughs in a wave. Taking `φ = 0` in Eq. (15.2), the displacement at `t = 0` is given by

`color{blue} {y(x,0) = a sin kx}`

.......................(15.5)

`color{blue} ✍️`Since the sine function repeats its value after every `2π` change in angle,

`color{ blue} { sinkx = sin(kx + 2 n pi ) = sin k ( x + ( 2n pi ) /k ) }`

`color{blue} ✍️`That is the displacements at points `x` and at

`color {purple} {x + ( 2n pi )/k}`

`color{blue} ✍️`are the same, where `n=1,2,3,...` The 1east distance between points with the same displacement (at any given instant of time) is obtained by taking `n = 1. lamda` is then given by

`color {blue} { lamda = (2 pi)/k }` or

`color {blue} { k (2 pi)/lamda}`

..................(15.6)

`color{blue} ✍️``k` is the angular wave number or propagation constant; its SI unit is radian per metre or ` rad m^-1`

`color {brown}bbul " Period, Angular Frequency and Frequency"`
`color{blue} ✍️`Fig. 15.7 shows again a sinusoidal plot. It describes not the shape of the wave at a certain instant but the displacement of an element (at any fixed location) of the medium as a function of time.

`color{blue} ✍️`We may for, simplicity, take Eq. (15.2) with `φ = 0` and monitor the motion of the element say at `x = 0` . We then get

`color {blue} {y(0,t ) = a sin(- omega t )
= - a sin omega t}`

`color{blue} ✍️`Now the period of oscillation of the wave is the time it takes for an element to complete one full oscillation. That is

`color{purple} {- a sinomega t = -a sin omega (t + T)}`

`color{purple} { = - a sin( omega t + omega T)}`

`color{blue} ✍️`Since sine function repeats after every `2 pi`,

`color{blue} {omega T = 2 pi " or " omega = (2pi)/T}`

.........................(15.7)

`omega` is called the angular frequency of the wave.

`color{blue} ✍️`Its SI units is `rad s^-1`. The frequency `ν` is the number of oscillations per second. Therefore,

`color{blue} {v = 1/T = omega/(2 pi)}`

........................... (15.8)

`v` is usually measured in hertz.

`color{blue} ✍️`In the discussion above, reference has always been made to a wave travelling along a string or a transverse wave. In a longitudinal wave, the displacement of an element of the medium is parallel to the direction of propagation of the wave. In Eq. (15.2), the displacement function for a longitudinal wave is written as,

`color{blue} {s(x, t) = a sin (kx – ωt + φ )} `

...............................(15.9)

`color{blue} ✍️`where `s(x, t)` is the displacement of an element of the medium in the direction of propagation of the wave at position x and time `t`. In Eq. (15.9), a is the displacement amplitude; other quantities have the same meaning as in case of a transverse wave except that the displacement function `y (x, t )` is to be replaced by the function `s (x, t).`